Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138832 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5,7,8)(4,6)(9,10,12,18,22,24,21,15,14,20,11,17,13,19,23,16);(1,3,4,7)(2,5)(6,8)(10,15)(11,21)(12,23)(13,22)(16,20)(17,19)(18,24);(10,16)(11,21)(12,23)(13,22)(15,20)(17,24)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72279467 |
| maximal normal subgroup of | #72279467 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46112);