#715101

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

1614 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fhi,d^2=h,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, 622297);