Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,8)(5,6)(9,10,12,16,22,21,24,19,13,17,23,15,14,11,18,20);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,21)(15,23)(17,22)(18,19)(20,24);(1,4)(2,3)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,f^c=fgi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219622, #72224569, #72224575, … |
| maximal normal subgroup of | #72219622, #72224569, #72224575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38479);