#6763621

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

1377 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,32)(27,29,34,37)(28,31,35,38)(33,36,39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,32,30,34,37,26)(28,33,36,38,35,39,40,31);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(28,35)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=f,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, 6670817);