#4512442

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1458 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4419638);