Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184773 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups; only the 330 normal subgroups are shown.
This group has 2022 conjugacy classes of subgroups — too many to draw. Only its 330 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 — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23,24,21)(20,22);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16);(1,4,5,3)(2,6,7,8)(10,14)(13,16);(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264285, #72264492 |
| maximal normal subgroup of | #72264285, #72264492 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299527);