#6344642

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=eh,f^a=fh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251838);