#735297

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(2,3)(6,7)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,36)(35,39);(10,13)(14,16)(25,26,29,30)(27,31,33,34)(28,36,37,32)(35,40,39,38);(1,2,5,3)(4,6,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,5,2)(4,7,8,6)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=hi,d^2=fi,e^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, 642493);