#640945

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

1582 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=i,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 548141);