Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14,15,18,12,17)(11,19,20,24,23,21,22,16);(1,3)(2,5,6,7)(4,8)(9,11,14,21,13,22,18,24,15,23,17,19,12,20,10,16);(9,10,13,14,15,18,12,17)(11,19,20,24,23,21,22,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280553, #72280614, C2×#128384 |
| maximal normal subgroup of | #72280553, #72280614, C2×#128384 |
| maximal quotient of | C2×#128384 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35580);