#733999

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

1572 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fgi,d^2=fi,e^2=gh,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, 641195);