#6808284

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	#30397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(6,9)(7,11)(13,14,18,23)(15,19,24,32)(16,21,27,35)(17,20,29,22)(25,31,37,42)(26,33,38,30)(28,34,40,36)(39,43,44,41);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,31)(22,30)(23,32)(27,37)(28,39)(29,38)(34,43)(35,42)(36,41)(40,44);(1,2,4,3)(14,20)(15,24)(17,29)(19,30)(21,34)(22,23)(25,37)(28,40)(31,41)(32,33)(35,36)(42,43);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(24,37)(26,39)(29,40)(30,41)(32,42)(33,43)(38,44);(1,3,4,2)(6,9)(10,12)(15,24)(16,27)(19,32)(21,35)(26,38)(28,40)(30,33)(34,36)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,e^2=hi,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, 6715480);