#6808150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715346);