Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129767 |
| 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 | #60444 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 972 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,7,8,5)(4,6)(10,14)(11,17)(12,20)(15,22)(16,23)(18,21)(19,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,21)(13,15)(14,17)(16,20)(19,23)(22,24);(11,19)(14,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219612, #72224569, #72224579, … |
| maximal normal subgroup of | #72219612, #72224569, #72224579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36963);