#697174

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 604370);