#72285416

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

3280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,7)(4,5)(8,11)(15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,2,6,7)(3,11,13,8)(4,9,5,10)(12,16,14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31);(1,3)(2,8,7,11)(4,12)(5,14)(6,13)(9,15,10,16)(17,19)(18,22,24,25)(20,29)(21,27)(23,31,28,32)(26,30);(1,4)(2,9,7,10)(3,12)(5,6)(8,15,11,16)(13,14)(17,19)(18,22,24,25)(20,26)(21,27)(23,31,28,32)(29,30);(1,5,6,4)(2,10)(3,12,13,14)(7,9)(8,16)(11,15)(17,20)(18,23,24,28)(19,26)(21,30)(22,31,25,32)(27,29)

Polycyclic presentation:

2^10|b^2=ij,c^2=gh,d^2=gh,e^2=ghij,b^a=bij,c^a=cf,c^b=cg,d^a=dgij,d^b=dfij,e^a=egij,e^b=efgi,e^c=eghi,e^d=eij,f^c=fj,f^d=fi,f^e=fij,g^c=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72310940, #72311155, …
maximal normal subgroup of	#72285526, #72310940, #72311155, …

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