#768726

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

1510 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)(7,8)(18,21)(19,23)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(2,6)(7,8)(9,10,13,16)(11,14,12,15)(26,30)(31,33)(32,36)(39,40);(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,18,20,21)(19,22,23,24)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,15,16,14)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=g,d^2=h,e^2=fh,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, 675922);