#4397617

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

1446 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)(3,4)(5,7)(6,8)(11,15)(13,16)(17,18,22,19)(20,24,21,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,5,8)(2,6,7,4)(17,19,22,18)(20,24,21,23)(26,30)(27,33)(32,35)(34,39);(2,7)(4,6)(9,10,12,14)(11,13,15,16)(17,20,22,21)(18,23,19,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,21,22,20)(18,24,19,23)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4)(2,3)(5,6)(7,8)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=i,d^2=hi,e^2=h,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, 4304813);