Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364611 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2319 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)(3,13)(4,18,22,9)(5,23)(6,26)(7,12)(8,16)(10,24)(11,29)(14,41,19,39)(15,31)(17,37)(20,51,53,56)(21,35,55,48)(25,38)(27,32)(28,36)(30,43,33,50)(34,52,47,54)(40,44)(42,60,49,57)(45,59)(46,58)(61,63,62,64);(1,3)(2,8)(4,19,22,14)(5,15)(6,27)(7,17)(9,33,18,30)(10,31)(11,37)(12,32)(13,26)(16,29)(20,49,53,42)(21,43,55,50)(23,40)(24,46)(25,45)(28,44)(34,60,47,57)(35,39,48,41)(36,59)(38,58)(51,61,56,62)(52,63,54,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,30)(10,34)(11,35)(12,18)(13,39)(15,42)(16,43)(17,19)(23,56)(24,52)(25,53)(26,48)(27,50)(28,54)(29,55)(31,57)(32,33)(36,47)(37,41)(38,51)(40,61)(44,63)(45,49)(46,64)(58,62)(59,60);(1,5,7,25)(2,10,12,36)(3,15,17,45)(4,20,22,53)(6,28,29,24)(8,31,32,59)(9,34,18,47)(11,38,26,23)(13,40,37,58)(14,42,19,49)(16,46,27,44)(21,54,55,52)(30,57,33,60)(35,51,48,56)(39,61,41,62)(43,64,50,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271808);