Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C22 | #1316 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(5,6,9,15,11,16,29,46,12,17,30,47,10,31,34,45)(7,18,20,48,21,49,27,61,22,50,56,64,32,57,55,68)(8,23,24,58,25,14,40,41,26,42,28,62,33,43,60,67)(13,35,36,54,38,63,65,52,39,59,66,19,37,51,44,53);(1,2,3,4)(5,7)(6,13)(8,19)(9,27)(10,32)(11,21)(12,22)(14,18)(15,44)(16,38)(17,39)(20,34)(23,57)(24,59)(25,53)(26,54)(28,35)(29,56)(30,55)(31,37)(33,52)(36,46)(40,51)(41,61)(42,49)(43,50)(45,66)(47,65)(48,58)(60,63)(62,64)(67,68)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,f^a=fh,h^a=hi,d^b=dfi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1290);