Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,16)(4,21)(5,9)(6,15)(7,13)(8,14)(10,19)(11,22)(12,25)(17,45)(18,43)(20,41)(23,56)(24,54)(26,42)(27,33)(28,32)(29,37)(30,40)(31,50)(34,58)(35,60)(36,52)(38,57)(39,55)(44,46)(47,53)(48,59)(49,63)(51,61)(62,64);(2,9)(5,25)(6,7)(10,34)(11,21)(12,15)(13,33)(14,32)(16,43)(18,50)(19,29)(22,26)(23,30)(27,28)(31,37)(35,59)(36,41)(38,56)(39,54)(40,42)(44,45)(46,51)(47,48)(52,58)(53,61)(55,57)(60,64)(62,63);(1,3,8,20)(2,10,14,37)(4,17,24,49)(5,18,28,52)(6,29,7,19)(9,31,32,34)(11,35,39,61)(12,36,33,43)(13,41,15,16)(21,53,54,59)(22,46,55,64)(23,48,30,47)(25,50,27,58)(26,51,57,60)(38,62,42,44)(40,63,56,45);(1,4)(2,11)(3,17)(5,26)(6,23)(7,30)(8,24)(9,21)(10,35)(12,40)(13,38)(14,39)(15,42)(16,44)(18,51)(19,47)(20,49)(22,25)(27,55)(28,57)(29,48)(31,53)(32,54)(33,56)(34,59)(36,63)(37,61)(41,62)(43,45)(46,50)(52,60)(58,64)Polycyclic presentation:
2^9|c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250056, #72250066, #72250217, … |
| maximal normal subgroup of | #72250056, #72250066, #72250217, … |
| maximal quotient of | #72250557, #72280369, #72281756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406944);