#72285102

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

2301 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)(3,8)(4,9)(5,18)(6,11)(7,14)(10,21)(12,24)(13,23)(15,22)(16,26)(17,27)(19,25)(20,28)(29,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,50)(44,51)(45,52)(46,56)(47,54)(48,55)(49,57)(53,61)(58,64)(59,63)(60,62);(1,3)(2,7)(4,12)(5,19,21,13)(6,14)(8,11)(9,22)(10,25,18,23)(15,17)(16,30,20,29)(24,27)(26,32,28,31)(33,35,38,45)(34,39,42,52)(36,43)(37,48,49,44)(40,50)(41,55,57,51)(46,60)(47,59)(53,64)(54,63)(56,62)(58,61);(1,3)(2,8)(4,15,17,12)(5,13)(6,14)(7,11)(9,22,27,24)(10,25)(16,30,20,29)(18,23)(19,21)(26,31,28,32)(33,36)(34,40)(35,43,45,59)(37,46)(38,47)(39,50,52,63)(41,56)(42,54)(44,58,48,60)(49,61)(51,64,55,62)(53,57);(1,4)(2,9)(3,12)(5,20,21,16)(6,17)(7,22)(8,24)(10,28,18,26)(11,27)(13,30,19,29)(14,15)(23,32,25,31)(33,37)(34,41)(35,44)(36,46,47,61)(38,49)(39,51)(40,53,54,56)(42,57)(43,58,59,60)(45,48)(50,62,63,64)(52,55);(1,5)(2,10)(3,13)(4,16)(6,21)(7,23)(8,25)(9,26)(11,18)(12,29)(14,19)(15,30)(17,20)(22,31)(24,32)(27,28)(37,49)(39,52)(41,57)(46,61)(50,63)(51,55)(53,56)(62,64)

Polycyclic presentation:

2^10|b^2=h,c^2=fhj,d^2=hj,b^a=bi,c^b=cf,d^a=dhj,d^b=dj,d^c=dg,e^a=efhij,e^b=egj,e^d=efh,f^b=fj,f^c=fj,f^d=fi,f^e=fi,g^a=gj,g^b=gi,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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.8);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
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.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);