#6816130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2119 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6723326);