Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,19,24,13)(8,25)(9,31,34,45)(10,18)(11,17)(12,16)(14,28)(15,32,44,33)(20,39,49,62)(21,50)(22,37,23,38)(26,35,48,46)(27,40,58,65)(29,41)(30,42)(36,51)(43,52)(47,66)(53,59)(54,56,60,57)(55,63)(61,68)(64,67);(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,39)(15,43)(16,19)(17,38)(18,37)(20,29)(25,46)(27,55)(28,49)(30,48)(31,59)(32,52)(33,51)(34,53)(35,42)(36,44)(40,64)(41,62)(45,50)(47,57)(54,61)(56,68)(58,67)(60,66)(63,65);(2,4)(5,8,10,28,12,30,11,29)(6,14,17,25,16,41,18,42)(7,20,22,48,24,49,23,26)(9,27,32,56,34,58,33,57)(13,35,37,62,19,46,38,39)(15,40,45,54,44,65,31,60)(21,47,51,67,53,68,52,55)(36,61,50,64,43,66,59,63);(2,4)(5,9,12,34)(6,15)(7,21,24,53)(8,27,30,58)(10,32,11,33)(13,36)(14,40)(16,44)(17,45)(18,31)(19,43)(20,47,49,68)(22,51,23,52)(25,54)(26,55,48,67)(28,56,29,57)(35,61)(37,50)(38,59)(39,63)(41,65)(42,60)(46,66)(62,64)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293280);