#6849791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

1198 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=ghi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6756987);