Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22,24,27,25,28,33,36)(23,29,30,34,31,35,32,26);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20);(2,4)(5,8,9,6)(7,11,12,10)(14,18)(17,20);(7,12)(10,11)(21,23,25,31)(22,26,28,34)(24,32,33,30)(27,35,36,29)Polycyclic presentation:
2^9|a^2=eh,b^2=f,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357084);