#72338538

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	C₄	#51889
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#1419457
prime radical	#403997	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=h,d^2=j,e^2=ij,g^2=j,c^a=cf,c^b=cg,d^a=dgij,d^b=dgij,d^c=dgij,e^a=ehij,e^c=eghj,e^d=eghj,f^e=fj,g^c=gj,h^d=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338568
maximal normal subgroup of	#72338568

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