#6356005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1680 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263201);