Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| 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 | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,12,16,11,18,10,17)(7,19,23,36,24,43,22,13)(8,25,27,40,28,39,33,14)(9,29,32,42,31,41,30,15)(20,37,47,44,49,34,48,60)(21,50,52,63,53,35,51,45)(26,38,55,54,56,62,58,57)(46,65,66,61,68,64,67,59);(1,2,4,3)(5,7,11,24)(6,13)(8,20)(9,21,31,53)(10,22,12,23)(14,37,40,34)(15,35)(16,19)(17,43)(18,36)(25,44,39,60)(26,46)(27,47)(28,49)(29,45)(30,51,32,52)(33,48)(38,61,62,59)(41,63)(42,50)(54,64,57,65)(55,66)(56,68)(58,67);(5,8,12,33,11,28,10,27)(6,14,17,25,18,40,16,39)(7,20,23,48,24,49,22,47)(9,26,32,58,31,56,30,55)(13,34,43,60,36,37,19,44)(15,38,41,54,42,62,29,57)(21,46,52,67,53,68,51,66)(35,59,63,65,50,61,45,64);(1,3,4,2)(5,9,12,32,11,31,10,30)(6,15,17,41,18,42,16,29)(7,21,23,52,24,53,22,51)(8,26,33,58,28,56,27,55)(13,35,43,63,36,50,19,45)(14,38,25,54,40,62,39,57)(20,46,48,67,49,68,47,66)(34,59,60,65,37,61,44,64)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276390);