Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | #2163 |
| prime radical | C24 | M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,14,13)(15,24,28,26)(16,25,29,27)(30,34,36,35);(2,4)(5,6,8,15,9,16,20,30,10,12,21,24,22,25,33,34,11,14,19,28,23,29,32,36,7,13,17,26,18,27,31,35);(6,13,14,12)(8,19)(15,24,28,26)(16,27,29,25)(17,21)(20,32)(30,34,36,35)(31,33)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^b=ch,d^b=di,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72217183, #72217185, #72217188, … |
| maximal normal subgroup of | #72217183, #72217185, #72217188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59089);