#72237047

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#604823
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#6437449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5698 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fgj,c^2=i,g^2=j,h^2=j,c^a=cg,c^b=cghj,d^a=dghj,d^b=di,e^a=eghj,e^b=egi,e^c=eij,f^c=fj,g^e=gj,h^b=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237050, #72237061, #72237665, …
maximal normal subgroup of	#72237050, #72237061, #72237665, …

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