#6804668

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

1243 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)(25,26,30,34)(27,29,33,31)(28,36,38,35)(32,37,40,39);(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)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(2,5)(6,7)(25,28,29,37,30,38,31,39)(26,32,27,35,34,40,33,36);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,26)(27,29)(28,32)(30,34)(31,33)(35,37)(36,39)(38,40);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,36,40,35)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=fi,e^2=hi,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^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, 6711864);