Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,12,19,14,20,13,39)(7,21,25,43,26,52,40,42)(8,27,32,49,34,58,33,62)(9,35,10,37,36,18,11,17)(15,24,44,38,46,57,45,23)(16,47,50,31,51,29,61,30)(22,41,55,65,56,67,63,66)(28,53,59,48,60,68,54,64);(5,7)(6,15)(8,28)(9,24)(10,38)(11,23)(12,25)(13,40)(14,26)(16,48)(17,42)(18,52)(19,44)(20,46)(21,35)(22,31)(27,41)(29,55)(30,56)(32,59)(33,54)(34,60)(36,57)(37,43)(39,45)(47,63)(49,65)(50,68)(51,64)(53,61)(58,67)(62,66);(2,4)(5,8,9,29,12,32,10,30,14,34,36,47,13,33,11,31)(6,16,17,49,39,61,18,27,20,51,37,62,19,50,35,58)(7,22,23,54,40,63,57,60,26,56,38,59,25,55,24,28)(15,41,21,53,44,65,43,48,46,67,52,68,45,66,42,64)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55711);