#6693067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6600263);