#658513

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=h,d^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, 565709);