#6911715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6818911);