#732394

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

1706 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=i,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, 639590);