#679583

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1792 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)(4,6)(5,8)(9,10,13,16)(11,12,15,14)(17,18,21,23)(19,20,24,22)(25,26,29,28)(27,31,32,30);(1,3,5,4)(2,6,8,7)(17,19,21,24)(18,22,23,20)(26,28)(30,31);(1,3,5,4)(2,7,8,6)(9,11,13,15)(10,14,16,12)(17,18)(19,20)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,3)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,7)(4,6)(5,8)(9,12,13,14)(10,15,16,11)(17,19)(18,20)(21,24)(22,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=fg,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 586779);