Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,18,35,17)(7,19,22,38,24,50,23,58)(8,25,28,45,29,14,41,44)(9,15,31,46,32,47,10,33)(13,36,39,57,40,34,48,21)(20,51,54,65,56,49,55,67)(26,43,27,60,30,42,62,63)(37,61,66,53,59,64,68,52);(1,2,4,3)(5,7)(6,13)(8,20)(9,21)(10,34)(11,22)(12,24)(14,37)(15,38)(16,39)(17,48)(18,40)(19,33)(23,35)(25,59)(26,52)(27,61)(28,54)(29,56)(30,53)(31,36)(32,57)(41,55)(42,65)(43,67)(44,66)(45,68)(46,50)(47,58)(49,63)(51,60)(62,64);(1,3,4,2)(5,8)(6,14)(7,20)(9,30)(10,27)(11,28)(12,29)(13,37)(15,43)(16,44)(17,45)(18,25)(19,49)(21,53)(22,54)(23,55)(24,56)(26,32)(31,62)(33,63)(34,61)(35,41)(36,64)(38,67)(39,66)(40,59)(42,47)(46,60)(48,68)(50,51)(52,57)(58,65);(5,9,11,31,12,32,35,10)(6,15,16,46,18,47,17,33)(7,21,22,36,24,57,23,34)(8,26,28,27,29,30,41,62)(13,38,39,50,40,58,48,19)(14,42,44,63,25,43,45,60)(20,52,54,61,56,53,55,64)(37,65,66,49,59,67,68,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=ci,e^a=eh,h^a=hi,e^b=eh,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, 400441);