Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387433 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2372 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14479 |
| 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)(4,20,25,47)(5,26,30,11)(6,31)(7,12)(8,13)(9,17)(10,24,38,23)(15,32,43,59)(16,39,45,53)(18,35)(19,34)(21,36,42,58)(22,49)(27,33,44,60)(28,40,29,55)(37,51)(41,57)(46,52)(48,61)(50,54)(56,64)(62,63);(1,3)(2,9)(4,21)(5,27)(6,18)(7,17)(8,19)(10,36)(11,39)(12,34)(13,35)(14,31)(15,24)(16,28)(20,32)(22,41)(23,43)(25,42)(26,53)(29,45)(30,44)(33,40)(37,62)(38,58)(46,61)(47,59)(48,50)(49,64)(51,56)(52,63)(54,57)(55,60);(1,4)(2,10,13,38)(3,15,19,43)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37,26,54)(12,20,31,47)(14,36)(16,41,45,63)(17,42,18,21)(27,56,44,48)(28,50)(29,51)(30,52)(33,57)(34,58)(35,59)(39,61)(40,46,55,49)(53,64)(60,62);(1,5,7,29,8,30,6,28)(2,11,12,40,13,26,31,55)(3,16,18,44,19,45,17,27)(4,22,24,51,25,52,23,50)(9,33,14,39,35,60,34,53)(10,37,20,46,38,54,47,49)(15,41,21,48,43,63,42,56)(32,57,36,61,59,62,58,64)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294751);