Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13717 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2233 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | C2×SD32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,3)(5,6)(7,17,20,47)(8,12)(9,25)(10,30)(11,33)(13,19,35,50)(14,28)(15,32)(16,18)(21,55,54,68)(22,56,34,48)(23,57)(24,41)(26,46,58,43)(27,37)(29,45)(31,40,52,67)(36,59)(38,63,66,65)(39,62,44,53)(42,51)(49,61)(60,64);(1,2,4,3)(6,12)(9,23)(13,35)(15,37)(16,33)(19,50)(26,58)(28,59)(29,49)(30,41)(38,66)(39,44)(43,46)(51,64)(53,62)(63,65);(1,3,4,2)(5,7,18,22,8,20,11,34)(6,13,33,39,12,35,16,44)(9,26,49,63,23,58,29,65)(10,31,27,54,24,52,32,21)(14,40,60,55,36,67,42,68)(15,43,41,66,37,46,30,38)(17,45,48,57,47,61,56,25)(19,51,53,59,50,64,62,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=dh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60021);