Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C23 | #508 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(5,6,10,17,31,54,29,64,12,18,9,16,11,15,38,48,13,19,35,51,33,49,34,47,28,46,36,52,8,27,30,65)(7,20,23,56,21,58,32,14,25,42,61,63,24,50,37,44,26,57,62,43,39,53,66,45,40,67,22,41,60,68,59,55);(1,2,3,4)(5,7)(6,14)(8,21)(9,32)(10,37)(11,39)(12,40)(13,26)(15,41)(16,50)(17,53)(18,55)(19,45)(20,47)(22,34)(23,38)(24,33)(25,28)(27,63)(29,61)(30,62)(31,60)(35,59)(36,66)(42,48)(43,49)(44,46)(51,58)(52,68)(54,56)(57,64)(65,67);(5,8,28,33,13,11,12,31)(6,15,46,54,19,27,18,49)(7,21,25,24,26,39,40,60)(9,29,10,30,36,34,35,38)(14,41,44,56,45,63,55,43)(16,47,17,48,52,64,51,65)(20,53,42,68,57,58,67,50)(22,59,23,32,61,37,62,66)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286491, #72286505, #72286519, … |
| maximal normal subgroup of | #72286491, #72286505, #72286519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59192);