#4409942

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fgh,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4317138);