Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1825 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11358 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,13,8,15,25,14)(3,16,21,47,22,39,20,49)(4,23,28,48,29,58,27,50)(5,30,33,11,6,34,37,12)(9,31,41,18,42,35,40,19)(10,32,45,26,46,36,44,59)(17,38,24,43,53,63,56,62)(51,54,57,52,64,55,61,60);(1,3)(2,9)(4,24,29,56)(5,31)(6,35)(7,20)(8,22)(10,43,46,62)(11,47)(12,49)(13,40)(14,41)(15,42)(16,34)(17,27,53,28)(18,37)(19,33)(21,25)(23,57,58,61)(26,52,59,60)(30,39)(32,55,36,54)(38,44,63,45)(48,51,50,64);(1,4,25,28,8,29,7,27)(2,10,13,44,15,46,14,45)(3,17,20,24,22,53,21,56)(5,32,37,26,6,36,33,59)(9,38,41,62,42,63,40,43)(11,48,34,23,12,50,30,58)(16,51,47,61,39,64,49,57)(18,54,31,52,19,55,35,60)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56544);