Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
527 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,5,8)(3,6,10,13)(4,7,11,15)(9,12,16,14)(17,18,21,25)(19,26,29,31)(20,23,30,24)(22,27,32,28)(33,34)(35,38)(36,39)(37,40)(41,46)(42,45)(43,47)(44,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,22)(20,27)(21,29)(23,26)(24,31)(25,32)(28,30)(33,35)(34,38)(36,41)(37,42)(39,46)(40,45)(43,48)(44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,23)(19,27)(20,30)(24,25)(26,31)(28,29)(33,36,37,43)(34,39,40,47)(35,41,42,48)(38,44,45,46)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,d^c=dh,e^c=eg,f^c=fg50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10084);