#4453147

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2490 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=fg,e^2=h,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, 4360343);