Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator | C2 | C22×C82 |
| socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1354 conjugacy classes of subgroups; only the 710 normal subgroups are shown.
This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 710 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)(3,7)(4,6)(5,8)(9,10,11,13,12,14,15,16);(1,3)(2,5)(4,8)(6,7)(17,18,19,21,20,22,23,24);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=be,d^a=de,c^b=ce272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91614);