Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,17,12,18,10,16)(7,13,20,29,21,30,9,15)(8,22,27,14,25,34,26,33)(19,32,24,36,35,31,23,28);(1,3,4,2)(5,7,11,20,12,21,10,9)(6,13,17,29,18,30,16,15)(8,23,27,19,25,24,26,35)(14,31,34,28,33,32,22,36);(5,8)(6,14)(7,19)(9,24)(10,27)(11,26)(12,25)(13,28)(15,32)(16,34)(17,22)(18,33)(20,23)(21,35)(29,31)(30,36);(5,9,10,21,12,20,11,7)(6,15,16,30,18,29,17,13)(8,24,27,35,25,23,26,19)(14,32,34,36,33,31,22,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278315);