#6400683

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#548	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,16,15,17,18,20,19);(5,7,10,6)(8,11,9,12);(2,4);(5,8,10,9)(6,11,7,12);(5,9,10,8)(6,11,7,12)(13,15,16,18,17,19,20,14)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307879);