#725939

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1922 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,d^2=fghi,e^2=fgh,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, 633135);