#6809406

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6716602);