#729084

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

1766 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=fghi,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, 636280);