Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33604 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 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)(5,6,11,17,12,19,34,18)(7,20,25,50,26,38,56,51)(8,14,30,44,31,46,63,45)(9,15,10,33,21,16,35,48)(13,36,39,57,41,23,40,24)(22,49,54,67,27,58,60,68)(28,43,61,64,32,47,29,62)(37,52,65,59,42,55,66,53);(5,7,21,36,11,25,35,57,12,26,9,23,34,56,10,24)(6,13,33,51,18,40,15,38,19,41,48,50,17,39,16,20)(8,27,28,55,30,60,61,53,31,22,32,52,63,54,29,59)(14,42,62,67,45,65,47,49,46,37,64,68,44,66,43,58);(2,4)(5,8)(6,14)(7,22)(9,32)(10,29)(11,30)(12,31)(13,37)(15,47)(16,43)(17,44)(18,45)(19,46)(20,49)(21,28)(23,55)(24,53)(25,54)(26,27)(33,62)(34,63)(35,61)(36,52)(38,58)(39,65)(40,66)(41,42)(48,64)(50,67)(51,68)(56,60)(57,59);(5,9,11,10,12,21,34,35)(6,15,17,33,19,16,18,48)(7,23,25,24,26,36,56,57)(8,28,30,61,31,32,63,29)(13,38,39,51,41,20,40,50)(14,43,44,64,46,47,45,62)(22,52,54,59,27,55,60,53)(37,49,65,67,42,58,66,68)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403080);