Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14484 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
741 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)(3,14,18,40)(4,10)(5,25,29,54)(6,12)(7,30)(8,13)(9,17,33,19)(11,27,39,28)(15,41,44,34)(16,32)(20,47,43,31)(21,49,52,64)(22,36)(23,53)(24,37)(26,38)(35,50,60,51)(42,59)(45,57)(46,61)(48,58)(55,62)(56,63);(1,3)(2,9)(4,20)(5,26)(6,19)(7,17)(8,18)(10,34)(11,38)(12,40)(13,33)(14,30)(15,23)(16,27)(21,42)(22,44)(24,43)(25,32)(28,45)(29,46)(31,36)(35,59)(37,41)(39,61)(47,53)(48,50)(49,58)(51,55)(52,63)(54,57)(56,60)(62,64);(1,4,6,22,8,24,7,23)(2,10,12,36,13,37,30,53)(3,15,17,43,18,44,19,20)(5,21,27,50,29,52,28,51)(9,31,14,34,33,47,40,41)(11,35,25,49,39,60,54,64)(16,42,26,55,45,63,46,48)(32,56,38,58,57,59,61,62);(1,5)(2,11,13,39)(3,16,18,45)(4,21)(6,27)(7,28)(8,29)(9,32)(10,35,37,60)(12,25,30,54)(14,38)(15,42,44,63)(17,26,19,46)(20,48,43,55)(22,50)(23,51)(24,52)(31,56)(33,57)(34,58)(36,49,53,64)(40,61)(41,62)(47,59)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294795);