#6407547

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

1538 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=dfi,e^c=ei,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, 6314743);