Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31000 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13)(3,15,19,10)(4,11)(5,9,8,12)(7,14,25,40)(16,43,45,35)(17,33)(18,34,28,29)(20,38)(21,37)(22,36)(23,30)(24,31,26,39)(27,41,50,32)(42,49)(44,59)(46,63)(47,60)(48,55)(51,62)(52,61)(53,57)(54,56)(58,64);(2,9)(3,16)(4,20)(5,8)(7,26)(10,32)(11,36)(12,13)(15,41)(17,46)(18,50)(19,45)(21,52)(22,23)(24,25)(27,28)(29,35)(31,39)(33,58)(34,43)(37,56)(42,59)(44,47)(48,64)(49,53)(55,57)(60,63)(61,62);(1,3)(2,10)(4,17)(5,18)(6,19)(7,27)(8,28)(9,29)(11,33)(12,34)(13,15)(14,32)(16,26)(20,47)(21,53)(22,49)(23,48)(24,45)(25,50)(30,55)(31,43)(35,39)(36,42)(37,57)(38,60)(40,41)(44,51)(46,52)(54,64)(56,58)(59,62)(61,63);(1,4)(2,11)(3,17)(5,23)(6,22)(7,21)(8,20)(9,30)(10,33)(12,38)(13,36)(14,37)(15,42)(16,44)(18,48)(19,49)(24,54)(25,52)(26,51)(27,53)(28,47)(29,55)(31,56)(32,57)(34,60)(35,59)(39,62)(40,61)(41,63)(43,58)(45,64)(46,50)Polycyclic presentation:
2^9|a^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416975);