Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,3,4)(5,6)(7,18)(8,14)(9,15)(10,28)(11,32)(12,17)(13,21)(16,30)(19,43)(20,44)(22,45)(23,38)(24,39)(25,52)(26,55)(27,42)(29,35)(31,57)(33,41)(34,48)(36,58)(37,49)(40,54)(46,64)(47,65)(50,67)(51,62)(53,60)(56,63)(59,66)(61,68);(5,7)(6,13)(8,19)(9,20)(10,29)(11,33)(12,22)(14,34)(15,35)(16,41)(17,37)(18,28)(21,30)(23,46)(24,47)(25,53)(26,56)(27,49)(31,58)(32,36)(38,59)(39,60)(40,63)(42,44)(43,52)(45,57)(48,54)(50,68)(51,66)(55,61)(62,65)(64,67);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,23)(13,34)(15,39)(16,40)(17,38)(18,43)(20,47)(21,48)(22,46)(27,51)(28,52)(29,53)(30,54)(31,50)(32,55)(33,56)(35,60)(36,61)(37,59)(41,63)(42,62)(44,65)(45,64)(49,66)(57,67)(58,68);(8,23)(14,38)(19,46)(24,26)(25,50)(34,59)(39,55)(40,62)(43,64)(47,56)(48,66)(51,54)(52,67)(53,68)(60,61)(63,65);(5,9,12,11)(6,15,17,32)(7,20,22,33)(8,24,23,26)(10,27,31,30)(13,35,37,36)(14,39,38,55)(16,28,42,57)(18,44,45,41)(19,47,46,56)(21,29,49,58)(25,51,50,54)(34,60,59,61)(40,52,62,67)(43,65,64,63)(48,53,66,68)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=hi,h^2=i,b^a=bf,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529692);