#6686937

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,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, 6594133);