#659642

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

1660 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=f,d^2=h,e^2=g,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, 566838);