#7034730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,27,31)(23,32,24,29)(25,30,26,35)(28,34,36,33);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,35,32,30);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24,27,23)(22,29,31,32)(25,33,26,34)(28,35,36,30);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,30,31,35)(23,33,24,34)(28,29,36,32)

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308532, #72308817
maximal normal subgroup of	#72308532, #72308817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941926);