#669443

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

1888 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fgi,d^2=fgh,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, 576639);