#6761501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=h,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668697);