#4512514

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 2097152
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

1462 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)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(2,5)(6,8)(11,16)(13,15)(17,19,20,23)(18,21,22,24)(26,30)(27,34)(28,37)(38,39);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,23)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fh,d^2=fi,e^2=h,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, 4419710);