#6823368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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	#40173	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fgi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=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, 6730564);