#6356002

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1742 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:

(1,2,6,9)(3,10,13,16)(4,8,5,14)(7,11,15,12)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10);(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(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=h,b^2=fh,d^2=h,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, 6263198);