Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,6,9,14,10,13,24,30,11,15,25,32,8,20,22,31)(7,16,18,33,17,12,26,27,19,29,23,35,21,28,34,36);(5,7)(6,12)(8,21)(9,23)(10,17)(11,19)(13,29)(14,27)(15,28)(16,20)(18,25)(22,26)(24,34)(30,35)(31,33)(32,36);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240000, #72240699, #72240702, … |
| maximal normal subgroup of | #72240000, #72240699, #72240702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36422);