Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1390 |
| commutator | C2×D4 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7187 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,26,45,8,12,27,43,6,28,31,44)(3,14,16,46,17,47,24,56,18,38,52,60,29,59,51,63)(4,19,21,53,22,10,37,39,23,41,25,57,30,40,55,64)(9,32,33,20,36,54,62,49,34,58,61,15,35,48,42,50);(1,3)(2,9,12,34)(4,20)(5,24)(6,29)(7,17)(8,18)(10,38,40,14)(11,42,43,62)(13,36,28,35)(15,23)(16,31)(19,47,41,59)(21,32)(22,49)(25,58)(26,52)(27,51)(30,50)(33,44,61,45)(37,54)(39,63,64,56)(46,53,60,57)(48,55)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1294);