#6693089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600285);