#4456220

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1438 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4363416);