Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451734 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,25)(7,11)(8,16)(12,28)(14,43)(15,42)(17,39)(18,41)(19,33)(20,40)(21,34)(22,51)(23,53)(24,37)(26,31)(27,35)(29,45)(30,46)(32,47)(36,52)(38,55)(44,62)(48,61)(49,60)(50,58)(54,59)(56,63)(57,64);(1,3)(2,8)(4,18)(5,22)(6,26)(7,17)(9,32)(10,36)(11,31)(12,39)(13,25)(14,21)(15,24)(16,28)(19,48)(20,45)(23,46)(27,47)(29,34)(30,37)(33,57)(35,43)(38,42)(40,41)(44,50)(49,63)(51,53)(52,55)(54,64)(56,58)(59,62)(60,61);(1,4,7,21)(2,9,11,34)(3,14)(5,19,24,50)(6,27,28,20)(8,29)(10,33,37,58)(12,40,25,35)(13,41)(15,44)(16,47)(17,18)(22,48)(23,54,55,49)(26,45)(30,56)(31,32)(36,57)(38,60,53,59)(39,43)(42,61)(46,64)(51,62)(52,63);(1,5)(2,10)(3,15,17,22)(4,19)(6,23)(7,24)(8,30,31,36)(9,33)(11,37)(12,38)(13,42,39,51)(14,44,18,48)(16,46,26,52)(20,49)(21,50)(25,53)(27,54)(28,55)(29,56,32,57)(34,58)(35,59)(40,60)(41,61,43,62)(45,63,47,64);(2,11)(8,31)(9,34)(10,37)(12,25)(13,39)(29,32)(30,36)(33,58)(35,40)(38,53)(41,43)(42,51)(56,57)(59,60)(61,62)Polycyclic presentation:
2^9|c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358931);