#610651

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2258 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)(6,8)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,22,23,24)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,23)(18,22,20,24)(25,27,28,31)(26,29,30,32);(9,11,13,12)(10,15,16,14)(17,20,21,18)(19,22,23,24)(27,31)(29,32);(2,5)(6,8)(9,12,13,11)(10,15,16,14)(26,30)(29,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517847);