Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,11,17,12,16,10,30)(7,13,21,35,20,36,31,34)(8,22,25,38,26,48,24,14)(9,15,29,40,27,39,28,53)(18,41,44,32,43,56,52,57)(19,33,46,60,45,58,54,59)(23,47,51,61,49,65,50,37)(42,62,64,55,63,68,66,67);(1,2,3,4)(5,7,10,21,12,20,11,31)(6,13,17,34,16,36,30,35)(8,18,24,44,26,43,25,52)(9,19,28,46,27,45,29,54)(14,32,22,41,38,57,48,56)(15,33,40,59,39,58,53,60)(23,42,50,64,49,63,51,66)(37,55,47,62,61,67,65,68);(5,8,11,25,12,26,10,24)(6,14,30,48,16,38,17,22)(7,18,31,52,20,43,21,44)(9,23,29,51,27,49,28,50)(13,32,35,56,36,57,34,41)(15,37,53,65,39,61,40,47)(19,42,54,66,45,63,46,64)(33,55,60,68,58,67,59,62);(5,9)(6,15)(7,19)(8,23)(10,28)(11,29)(12,27)(13,33)(14,37)(16,39)(17,40)(18,42)(20,45)(21,46)(22,47)(24,50)(25,51)(26,49)(30,53)(31,54)(32,55)(34,59)(35,60)(36,58)(38,61)(41,62)(43,63)(44,64)(48,65)(52,66)(56,68)(57,67);(9,27)(15,39)(19,45)(23,49)(28,29)(33,58)(37,61)(40,53)(42,63)(46,54)(47,65)(50,51)(55,67)(59,60)(62,68)(64,66)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,h^b=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279228);