#72341015

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,25)(15,29)(17,32)(18,27)(19,22)(20,26)(21,28)(23,31)(24,30)(33,34)(35,40)(36,49)(37,53)(38,43)(39,44)(41,52)(42,55)(45,60)(46,62)(47,63)(48,59)(50,56)(51,61)(54,57)(58,64);(1,3)(2,8)(4,5)(6,15,21,31)(7,16)(9,22)(10,25)(11,23,28,29)(12,13)(14,19)(17,20,30,18)(24,26,32,27)(33,35)(34,40)(36,50)(37,54,55,46)(38,47,51,64)(39,48)(41,60)(42,62,53,57)(43,63,61,58)(44,59)(45,52)(49,56);(1,4,7,14)(2,9,12,25)(3,5,16,19)(6,17)(8,22,13,10)(11,24)(15,18)(20,31)(21,30)(23,27)(26,29)(28,32)(33,36)(34,41)(35,45)(37,38,55,51)(39,52)(40,56)(42,61,53,43)(44,49)(46,64,54,47)(48,50)(57,63,62,58)(59,60);(1,5)(2,10)(3,14,16,4)(6,20)(7,19)(8,9,13,25)(11,27)(12,22)(15,30,31,17)(18,21)(23,24,29,32)(26,28)(33,37)(34,42,44,53)(35,46,48,54)(36,51)(38,52)(39,55)(40,57)(41,61,49,43)(45,47,50,64)(56,63)(58,60)(59,62);(1,6,7,21)(2,11,12,28)(3,15)(4,17)(5,18,19,20)(8,23)(9,24)(10,26,22,27)(13,29)(14,30)(16,31)(25,32)(33,38,39,51)(34,43)(35,47)(36,37,52,55)(40,58,59,63)(41,53)(42,49)(44,61)(45,46)(48,64)(50,54)(56,57,60,62)

Polycyclic presentation:

2^10|b^2=g,c^2=gi,d^2=fghj,e^2=fghi,b^a=bi,c^a=cgj,c^b=cgj,d^a=dg,d^b=dh,d^c=df,e^a=egh,e^b=efhj,e^c=ehj,e^d=efghij,f^a=fj,f^b=fi,f^c=fj,f^d=fi,f^e=fj,g^d=gj,g^e=gij,h^b=hj,h^c=hij,h^d=hij,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341033, #72341083
maximal normal subgroup of	#72341033, #72341083

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);