Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×C4×#120 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,25)(3,14,17,9,18,31,26,30)(4,10,20,34,21,35,27,33)(5,11,24,37,22,36,23,49)(15,40,44,53,43,38,42,28)(16,39,46,29,45,55,50,54)(19,32,48,58,47,56,51,57)(41,60,63,64,61,59,62,52);(1,3)(2,9)(4,15)(5,16)(6,26)(7,17)(8,18)(10,28)(11,29)(12,31)(13,30)(14,25)(19,41)(20,44)(21,43)(22,45)(23,50)(24,46)(27,42)(32,52)(33,38)(34,40)(35,53)(36,54)(37,55)(39,49)(47,61)(48,63)(51,62)(56,64)(57,59)(58,60);(1,4)(2,10)(3,15)(5,19)(6,27)(7,20)(8,21)(9,28)(11,32)(12,34)(13,35)(14,38)(16,41)(17,44)(18,43)(22,47)(23,51)(24,48)(25,33)(26,42)(29,52)(30,53)(31,40)(36,56)(37,58)(39,59)(45,61)(46,63)(49,57)(50,62)(54,64)(55,60);(1,5,7,24,8,22,6,23)(2,11,12,37,13,36,25,49)(3,16,17,46,18,45,26,50)(4,19,20,48,21,47,27,51)(9,29,31,55,30,54,14,39)(10,32,34,58,35,56,33,57)(15,41,44,63,43,61,42,62)(28,52,40,60,53,64,38,59);(5,22)(11,36)(16,45)(19,47)(23,24)(29,54)(32,56)(37,49)(39,55)(41,61)(46,50)(48,51)(52,64)(57,58)(59,60)(62,63)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,g^b=gi,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72312363, #72312375, #72312558, … |
| maximal normal subgroup of | #72312363, #72312375, #72312558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279918);