Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| 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 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)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,c^b=ci,d^b=dg,e^b=eg,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36862);