Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1466 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
510 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 510 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,16,12,15,11,10)(7,13,19,20,18,29,26,32)(8,14,22,23,25,31,24,27)(17,28,21,30,33,36,35,34);(5,7,10,20,11,19,15,29,12,18,16,32,9,26,6,13)(8,21,23,28,24,33,14,30,25,35,27,36,22,17,31,34);(1,3,4,2)(5,8,9,22,12,25,11,24)(6,14,16,23,15,31,10,27)(7,17,19,21,18,33,26,35)(13,28,20,30,29,36,32,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313521, #72313559 |
| maximal normal subgroup of | #72313521, #72313559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35576);