#72285258

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

2361 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fghj,d^2=hi,c^a=ch,c^b=cg,d^a=dhij,d^b=dfghj,d^c=dfgj,e^a=efhij,e^b=eghj,e^c=efhj,e^d=efhj,f^c=fi,f^d=fij,f^e=fij,g^a=gj,g^b=gi,g^c=gij,g^d=gi,g^e=gij,h^b=hj,h^c=hj,h^e=hij

Conjugacy classes & character table

55 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.6);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.10);;
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.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
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*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
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.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);