#6358162

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1548 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,13)(15,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3)(2,7)(4,12)(5,15)(6,13)(8,11)(9,16)(10,14);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,2)(3,11)(4,8)(5,14)(6,9)(7,15)(10,13)(12,16);(1,4,6,5)(2,8,9,14)(3,12,13,15)(7,11,16,10)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265358);