#6808937

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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
prime radical	#30560	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716133);