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 | |
1560 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)(7,17)(8,22,25,13)(9,26,29,39)(10,30)(11,16)(12,20)(14,28,40,47)(15,32)(18,43,46,42)(19,34,48,58)(21,41)(23,52)(24,37,57,51)(27,60,61,38)(31,35)(33,45,49,56)(36,54)(44,65)(50,66)(53,63)(55,59)(62,67)(64,68);(2,4)(5,7,11,21)(6,12,16,35)(8,18,25,46)(9,27,29,61)(10,31,32,20)(13,33,22,49)(14,38,40,60)(15,41,30,17)(19,47,48,28)(23,53,55,68)(24,56,57,45)(26,58,39,34)(36,63,50,64)(37,43,51,42)(44,66,67,54)(52,62,59,65);(1,2,3,4)(5,8,11,25)(6,13,16,22)(7,18,21,46)(9,23,29,55)(10,24,32,57)(12,33,35,49)(14,36,40,50)(15,37,30,51)(17,42,41,43)(19,44,48,67)(20,45,31,56)(26,59,39,52)(27,53,61,68)(28,54,47,66)(34,62,58,65)(38,63,60,64);(5,9,10,28,11,29,32,47)(6,14,15,39,16,40,30,26)(7,19,20,27,21,48,31,61)(8,23,24,54,25,55,57,66)(12,34,17,38,35,58,41,60)(13,36,37,52,22,50,51,59)(18,44,45,53,46,67,56,68)(33,62,42,63,49,65,43,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=e,e^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, 293209);