Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle | C23 | C23×C8 |
| prime radical | C2×#120 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1135 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 302 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)(3,4)(5,6)(7,8)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23);(2,4);(6,8);(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72287670, #72319372, #72319629, … |
| maximal normal subgroup of | #72287670, #72319372, #72319629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402865);