#6791736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,33)(27,35,37,31)(28,38,39,32)(30,40,36,34);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,30)(29,37)(32,40)(33,35)(34,38)(36,39);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,31)(27,30)(29,39)(32,34)(33,35)(36,37)(38,40);(10,16)(14,17)(15,22)(21,23)(25,28)(26,32)(27,36)(29,39)(30,37)(31,34)(33,38)(35,40);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,37)(28,39)(31,35)(32,38)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,e^2=h,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^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6698932);