Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | C22×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,14)(9,16)(10,28)(11,32)(12,15)(13,21)(17,30)(19,43)(20,45)(22,44)(23,40)(24,38)(25,52)(26,55)(27,41)(29,36)(31,57)(33,42)(34,48)(35,49)(37,58)(39,54)(46,65)(47,64)(50,67)(51,63)(53,59)(56,62)(60,68)(61,66);(5,7)(6,13)(8,19)(9,20)(10,29)(11,33)(12,22)(14,34)(15,35)(16,36)(17,42)(18,28)(21,30)(23,46)(24,47)(25,53)(26,56)(27,49)(31,58)(32,37)(38,59)(39,62)(40,61)(41,45)(43,52)(44,57)(48,54)(50,68)(51,66)(55,60)(63,64)(65,67);(1,3,4,2)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,23)(13,34)(15,40)(16,38)(17,39)(18,43)(20,47)(21,48)(22,46)(27,51)(28,52)(29,53)(30,54)(31,50)(32,55)(33,56)(35,61)(36,59)(37,60)(41,63)(42,62)(44,65)(45,64)(49,66)(57,67)(58,68);(6,15)(8,23)(13,35)(16,32)(17,41)(18,44)(19,46)(24,26)(25,50)(28,57)(36,37)(42,45)(47,56)(48,66)(51,54)(53,68);(5,9,12,11)(6,16,15,32)(7,20,22,33)(8,24,23,26)(10,27,31,30)(13,36,35,37)(14,38,40,55)(17,28,41,57)(18,45,44,42)(19,47,46,56)(21,29,49,58)(25,51,50,54)(34,59,61,60)(39,52,63,67)(43,64,65,62)(48,53,66,68)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=i,f^2=hi,h^2=i,b^a=bf,d^a=di,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, 7529807);