Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| 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 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,31)(7,18,21,13,22,38,32,37)(8,23,25,40,26,14,33,41)(9,27,29,44,30,43,34,15)(19,46,49,35,50,45,48,53)(20,36,51,42,52,60,28,62)(24,55,58,54,59,61,57,39)(47,66,56,64,67,63,68,65);(2,4)(5,7)(6,13)(8,19)(9,28)(10,32)(11,21)(12,22)(14,35)(15,42)(16,38)(17,37)(18,31)(20,29)(23,53)(24,56)(25,49)(26,50)(27,60)(30,51)(33,48)(34,52)(36,43)(39,64)(40,46)(41,45)(44,62)(47,57)(54,65)(55,63)(58,67)(59,68)(61,66);(1,2,3,4)(5,8,12,26)(6,14,17,23)(7,19,22,50)(9,24,30,59)(10,25,11,33)(13,35,37,53)(15,39,44,54)(16,40,31,41)(18,45,38,46)(20,47,52,67)(21,48,32,49)(27,61,43,55)(28,56,51,68)(29,57,34,58)(36,63,60,66)(42,64,62,65);(5,9,10,29,12,30,11,34)(6,15,16,43,17,44,31,27)(7,20,21,28,22,52,32,51)(8,24,25,57,26,59,33,58)(13,36,18,42,37,60,38,62)(14,39,40,55,23,54,41,61)(19,47,48,56,50,67,49,68)(35,63,45,64,53,66,46,65)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293211);