#878453

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

1576 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)(9,10,13,16)(11,12,15,14)(17,18,23,20)(19,21,24,22);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,12,15,14)(17,19)(18,22)(20,21)(23,24)(25,26,29,28)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,16,12)(17,20,23,18)(19,21,24,22)(25,27,29,32)(26,30,28,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21,23,22)(18,19,20,24)(25,28,29,26)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,15,16,11)(17,22,23,21)(18,24,20,19)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,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, 785649);