Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142406 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 70 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)(7,17)(8,22)(9,10)(13,29)(14,27)(15,23)(16,28)(18,26)(19,35)(20,21)(24,25)(30,31)(32,34);(5,6,9,14,11,16,10,15)(7,18,17,32,21,31,20,13)(8,23,22,28,24,27,25,12)(19,26,33,29,35,30,36,34);(2,4)(5,7,9,17,11,21,10,20)(6,13,15,31,16,32,14,18)(8,19,22,33,24,35,25,36)(12,26,27,34,28,30,23,29)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250331, #72250333, #72250336, … |
| maximal normal subgroup of | #72250331, #72250333, #72250336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49602);