#4320322

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1842 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:

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

Polycyclic presentation:

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

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, 4227518);