#6831247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,31)(27,32,36,40)(28,35,37,33)(29,34,39,38);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,36)(26,32,31,40)(28,29,37,39)(33,38,35,34);(2,5)(6,8)(25,28,31,35,30,37,26,33)(27,34,32,29,36,38,40,39);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,26,30,31)(27,32,36,40)(28,33,37,35)(29,38,39,34);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,29,26,34,30,39,31,38)(27,35,40,28,36,33,32,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6738443);