Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1467 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(2,4)(5,6,11,16,10,15,13,29,12,7,17,21,9,20,18,30)(8,22,26,35,25,19,34,33,27,23,28,36,24,14,31,32);(5,7,18,29,10,20,11,21,12,6,13,30,9,15,17,16)(8,23,31,33,25,14,26,36,27,22,34,32,24,19,28,35);(1,2,3,4)(5,8,10,25,12,27,9,24)(6,14,20,23,7,19,15,22)(11,28,13,31,17,26,18,34)(16,33,30,35,21,32,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,f^b=fgi,g^b=gi,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47899);