#1335720

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1242916);