Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #46225 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
957 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 957 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14,15,16)(18,21,22,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(9,10)(11,14)(12,16)(13,15)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72228001, #72228007, #72228013 |
| maximal normal subgroup of | #72228001, #72228007, #72228013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4512);