#6811514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(5,6)(7,10)(8,12)(9,11);(5,7,10,6)(8,12,9,11)(13,14,18,22)(15,23,27,38)(16,28,29,42)(17,31,24,21)(19,32,34,26)(20,35,36,30)(25,39,40,33)(37,43,44,41);(2,4)(13,15,24,32,18,27,17,26)(14,19,31,38,22,34,21,23)(16,25,35,43,29,40,30,41)(20,33,42,44,36,39,28,37);(5,8,10,9)(6,11,7,12)(16,29)(20,36)(25,40)(28,42)(30,35)(33,39)(37,44)(41,43);(5,9,10,8)(6,11,7,12)(13,16,17,30,18,29,24,35)(14,20,21,28,22,36,31,42)(15,25,26,41,27,40,32,43)(19,33,23,37,34,39,38,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=f,e^2=fh,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, 6718710);