#6808319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#30413	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(15,19,25,32)(16,21,27,35)(17,20,29,22)(24,31,38,43)(26,30,39,33)(28,34,40,36)(37,41,44,42);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,23,32)(16,24,27,38)(17,26,29,39)(20,30,22,33)(21,31,35,43)(28,37,40,44)(34,41,36,42);(1,2,4,3)(14,20)(17,29)(19,30)(21,34)(22,23)(26,39)(28,40)(31,41)(32,33)(35,36)(37,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(25,38)(26,37)(29,40)(30,41)(32,43)(33,42)(39,44);(1,3,4,2)(6,9)(10,12)(15,25)(16,27)(19,32)(21,35)(26,39)(28,40)(30,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715515);