Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511741 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24×D8 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4270 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 4270 conjugacy classes of subgroups — too many to draw. Only its 258 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 — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(2,4);(6,8)Polycyclic presentation:
2^9|a^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72231322, #72286189, C2×#511744, … |
| maximal normal subgroup of | #72231322, #72286189, C2×#511744, … |
| maximal quotient of | #72286620, #72286638, #72286659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418940);