Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #122971 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| 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 | |
342 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 342 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)(3,7)(4,6)(5,8)(9,13)(10,15)(11,16)(12,14)(17,18,23,27)(19,28,31,47)(20,25,22,37)(21,33,36,45)(24,39,42,30)(26,32,44,35)(29,46,38,43)(34,40,48,41);(2,6)(7,13)(8,14)(15,16)(17,19,23,31)(18,24,27,42)(20,29,22,38)(21,34,36,48)(25,40,37,41)(26,43,44,46)(28,45,47,33)(30,32,39,35);(1,3,5,10)(2,7,8,15)(4,11,12,9)(6,13,14,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,42,41)(26,33,44,45)(28,46,47,43)(30,34,39,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,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^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30186);