#6360943

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360896
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1518 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:

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

Polycyclic presentation:

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

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, 6268139);