#659179

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1730 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)(4,8)(5,6)(11,15)(12,16)(17,18)(19,21)(20,22)(23,24)(25,26,29,28)(27,31,32,30);(2,6)(3,8)(9,10,14,13)(11,12,15,16)(18,19)(20,24)(27,32)(30,31);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,31)(29,32);(1,4,5,7)(2,8,6,3)(9,12,14,16)(10,15,13,11);(1,2)(3,7)(4,8)(5,6)(9,13,14,10)(11,16,15,12)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,d^2=gh,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, 566375);