#740196

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

1728 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)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,20)(19,22,24,23)(26,30)(31,32);(1,2,5,3)(4,7,8,6)(10,14)(12,15)(18,20)(22,23);(2,3)(6,7)(9,11,13,16)(10,12,14,15)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,32)(29,30);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23)(25,27,29,28)(26,31,30,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=hi,d^2=fgi,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, 647392);