#6631925

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631112
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,d^c=dfh,g^c=gi

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, 6539121);