Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#2010 | 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 | |
798 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 798 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)(5,6,10,14)(7,15,18,33)(8,19,21,35)(9,24,22,13)(11,26,28,16)(12,29,31,23)(17,34,25,30)(20,32,36,27);(5,7)(6,11)(8,20)(9,25)(10,18)(12,30)(13,32)(14,28)(15,19)(16,23)(17,22)(21,36)(24,27)(26,29)(31,34)(33,35);(2,4)(8,21)(12,31)(19,35)(23,29)Polycyclic presentation:
2^9|a^2=i,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,g^b=gi,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213784, #72250353, #72250365, … |
| maximal normal subgroup of | #72213784, #72250353, #72250365, … |
| maximal quotient of | #72213261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49453);