#735639

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

1570 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,d^2=ghi,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, 642835);