Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C22 | #1352 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 68 points:
(1,2,3,4)(5,6,9,14,11,17,26,16)(7,18,20,44,23,46,22,55)(8,24,27,13,29,37,58,39)(10,31,30,15,32,40,56,42)(12,33,34,25,36,49,43,19)(21,51,50,45,54,67,53,66)(28,38,59,61,47,41,62,57)(35,48,64,63,65,52,60,68);(5,7,10,21,11,23,32,54)(6,12,15,35,17,36,42,65)(8,25,28,52,29,19,47,48)(9,20,30,50,26,22,56,53)(13,18,38,45,39,46,41,66)(14,34,40,64,16,43,31,60)(24,55,57,51,37,44,61,67)(27,49,59,68,58,33,62,63)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=eh,g^c=gi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1514);