Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #106 |
| commutator | C8 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle, prime radical | C23 | #340 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,13,12,14,15,16)(18,23,24,25)(19,27)(20,29)(21,28)(26,32,30,31);(2,5,7,6)(17,18,20,25,21,26,29,32,22,24,19,23,28,30,27,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216549, C8×#1368, #72223574, … |
| maximal normal subgroup of | #72216549, C8×#1368, #72223574, … |
| maximal quotient of | #72216542, C2×#93269, #72223652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 465);