#769295

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

1496 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)(10,13)(14,16)(17,18)(19,21)(20,23)(22,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,40,36,39);(17,19,22,23)(18,21,24,20)(26,30)(31,38)(32,35)(39,40);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,3)(2,7)(4,6)(5,8)(17,20,22,21)(18,23,24,19)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,19,24,23)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40)

Polycyclic presentation:

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

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, 676491);