#6407620

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=df,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314816);