#6407745

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,18,22,25,21,26,20,30)(19,23,29,31,28,24,27,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,31)(30,32);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20,21,22)(18,25,26,30)(19,27,28,29)(23,31,24,32);(18,26)(19,28)(25,30)(27,29)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfgi,d^b=df,e^b=ei,e^c=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, 6314941);