#6802431

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6709627);