#726178

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

1658 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)(3,7)(4,5)(6,8)(11,15)(14,16)(17,18,21,20)(19,23,24,22)(26,30)(31,32);(2,5)(6,7)(9,10,13,12)(11,14,15,16)(18,20)(22,23);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,16)(13,15)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,22,20,23)(25,27,29,28)(26,31,30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=gi,e^2=fg,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, 633374);