#729088

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

1790 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)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18,22,23)(19,20,21,24);(1,2,5,6)(3,4,7,8)(9,11)(10,12)(13,14)(15,16)(18,23)(19,21)(26,30)(31,32);(2,6)(3,7)(11,14)(12,16)(17,19,22,21)(18,24,23,20)(25,26)(27,31)(28,32)(29,30);(1,3)(2,4)(5,7)(6,8)(9,12,13,16)(10,14,15,11)(17,20,22,24)(18,19,23,21)(25,27,29,28)(26,31,30,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,21,22,19)(18,24,23,20)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,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, 636284);