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 | |
1252 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)(7,17,21,26)(8,22,25,14)(9,12)(10,15)(11,16)(13,20,29,19)(18,32)(23,30,24,27)(28,34)(31,35)(33,36);(6,12,16,15)(13,26,29,17)(14,27,22,30)(28,32,36,31);(5,7,9,19,11,21,10,20)(6,13,15,17,16,29,12,26)(8,18,23,33,25,35,24,34)(14,28,30,31,22,36,27,32);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,27)(13,28)(15,30)(16,22)(17,31)(19,33)(20,34)(21,35)(26,32)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,h^c=hi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299545);