Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,34)(25,29)(26,28)(27,32)(31,36)(33,35);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,32)(22,27,29,30)(24,33,26,31)(28,35,34,36);(5,7)(6,10)(8,11)(9,12)(14,18)(17,20)(23,31)(24,26)(27,35)(28,34)(30,36)(32,33);(6,9)(7,11)(15,19)(17,20)(21,24,25,26)(22,28,29,34)(23,31,32,33)(27,36,30,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298436);