#6651612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632036
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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,29,33)(27,30,34,38)(28,31,36,32)(35,39,40,37);(2,5)(3,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,37)(33,38)(36,40);(1,3,4,7)(2,6,5,8)(26,31)(27,34)(28,36)(30,37)(32,33)(38,39);(2,5)(6,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24);(10,16)(14,17)(15,20)(23,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=hi,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi

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, 6558808);