#6435883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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:

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

Polycyclic presentation:

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

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, 6343079);