Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1331 |
| prime radical | C23 | #359 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,6,9,16,12,19,11,39,13,21,35,18,40,20,41,48)(7,22,24,54,27,55,58,68,28,47,34,64,63,56,26,62)(8,29,31,51,10,36,32,15,33,49,37,50,38,65,61,17)(14,42,43,25,44,59,53,23,46,57,52,60,45,66,67,30);(2,4)(5,7,10,25,35,24,61,59,12,27,33,23,41,58,31,57,13,28,38,60,9,34,32,66,40,63,8,30,11,26,37,42)(6,14,17,22,48,53,65,68,20,45,50,56,18,43,49,54,21,46,15,47,39,67,36,62,19,44,51,55,16,52,29,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286505, #72286771, #72286915, … |
| maximal normal subgroup of | #72286505, #72286771, #72286915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1251);