#696663

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

1626 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=f,d^2=g,e^2=fg,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, 603859);