Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123131 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×C4⋊C8 | C23 |
| prime radical | #12158 | C2 |
| LCS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
377 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 44 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)(3,5)(4,7)(6,8)(10,14)(11,17)(12,20)(15,22)(16,23)(18,21)(19,24);(3,8)(9,10)(11,18)(12,21)(13,15)(14,17)(16,20)(19,23)(22,24);(2,5,7,6)(11,19)(14,22)(17,24)(18,23)Polycyclic presentation:
2^9|c^2=gh,d^2=fh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30329);