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 | #1343 |
| 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)(7,17)(8,22)(9,28)(10,31)(11,16)(12,18)(13,26)(14,29)(15,24)(19,47)(20,51)(21,43)(23,35)(25,36)(27,54)(30,60)(32,42)(33,62)(34,55)(37,56)(38,44)(39,52)(40,46)(41,61)(45,63)(48,58)(49,65)(50,66)(53,67)(57,59)(64,68);(5,7,9,19,11,21,30,50)(6,12,14,33,16,35,39,64)(8,23,25,48,27,18,44,45)(10,32,29,61,24,20,52,49)(13,17,36,41,37,43,38,65)(15,40,28,58,31,34,60,63)(22,51,53,47,54,42,59,66)(26,46,57,62,56,55,67,68)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,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, 1512);