#4418766

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3282 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)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,29,31,27)(26,32,28,34)(30,35,38,37)(33,40,36,39);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4325962);