Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #537 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,21,27,23,26,36,42,22,28,35,43,20,25,34,41)(19,29,32,44,31,24,37,38,33,40,46,48,30,39,45,47);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,29)(26,40)(27,38)(28,39)(34,45)(35,46)(36,37)(41,44)(42,48)(43,47);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,22,23)(18,25,28,26)(19,30,33,31)(21,34,35,36)(24,29,39,40)(27,41,43,42)(32,45,46,37)(38,44,47,48)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36864);