Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123211 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
271 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 271 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,8)(5,9)(6,11)(7,13)(12,14)(15,16)(17,18,22,26)(19,27,29,46)(20,31,33,45)(21,36,34,25)(23,38,39,28)(24,41,43,35)(30,47,37,42)(32,44,48,40);(1,3,5,12)(2,6,9,15)(4,11,13,16)(7,14,8,10)(17,19,22,29)(18,23,26,39)(20,32,33,48)(21,37,34,30)(24,42,43,47)(25,44,36,40)(27,45,46,31)(28,41,38,35);(1,4,5,13)(2,7)(3,11,12,16)(6,14)(8,9)(10,15)(20,33)(24,43)(27,46)(28,38)(31,45)(32,48)(35,41)(42,47)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ei,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30436);