#610652

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2112 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=hi,d^2=f,e^2=g,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, 517848);