#878403

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1652 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)(3,5,7,8)(9,10,13,15)(11,12,16,14)(17,18,22,20)(19,24,23,21)(26,31)(28,29);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(17,19)(18,21)(20,24)(22,23)(25,26)(27,28)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,18)(19,24,23,21)(25,27,30,32)(26,28,31,29);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(25,28,30,29)(26,27,31,32);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,24)(18,23,20,19)(25,29,30,28)(26,32,31,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 785599);