#686291

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593487);