Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,21,34)(8,13,23,30)(9,14)(10,15)(11,16)(12,20,28,35)(18,32)(19,33)(22,29,36,27)(24,31)(25,26);(5,7,18,35,11,21,10,20)(6,12,15,17,16,28,32,34)(8,19,36,25,23,24,22,9)(13,26,29,33,30,14,27,31);(5,8)(6,13)(7,19)(9,20)(10,22)(11,23)(12,26)(14,28)(15,29)(16,30)(17,33)(18,36)(21,24)(25,35)(27,32)(31,34);(2,4)(5,9,10,24,11,25,18,19)(6,14,15,31,16,26,32,33)(7,8,20,22,21,23,35,36)(12,27,17,13,28,29,34,30)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299599);