Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | #466 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,6,8,11)(3,9,10,12)(17,18)(19,27,29,44)(20,23)(21,33)(22,36)(24,28,38,46)(25,35)(26,32)(30,41,37,45)(31,43)(34,42,48,40)(39,47);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(18,23)(21,31)(24,38)(26,36)(28,46)(34,48)(35,47)(40,42);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,19)(18,24)(20,29)(21,34)(22,37)(23,38)(25,41)(26,42)(27,43)(28,47)(30,32)(31,48)(33,44)(35,46)(36,40)(39,45)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=g,e^2=gh,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44686);