#6686939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1909 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6594135);