Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129435 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2273 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1137 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17)(12,20,21,14)(13,16)(15,19)(18,22,24,23);(2,6)(3,4)(10,14)(15,22)(16,20)(17,23);(1,3,5,4)(2,7,6,8)(9,11)(10,15,16,17)(12,18)(13,19)(14,22,20,23)(21,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,f^b=fg,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36631);