Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142308 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 72 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)(5,6,12,18)(7,19,22,36)(8,24,9,28)(10,17,11,16)(13,30,32,21)(14,26,15,27)(20,33,25,35)(23,31,29,34);(5,7)(6,13)(8,25)(9,20)(10,29)(11,23)(12,22)(14,34)(15,31)(16,35)(17,33)(18,32)(19,24)(21,27)(26,30)(28,36);(2,4)(7,20)(8,26)(9,27)(10,11)(13,31)(14,28)(15,24)(16,17)(19,35)(21,29)(22,25)(23,30)(32,34)(33,36)Polycyclic presentation:
2^9|a^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50010);