Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| 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 | #33733 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,6,8,11)(3,9,10,12)(17,18,20,25,22,27,34,28)(19,29,30,42,32,44,33,23)(21,36,35,26,39,24,38,43)(31,45,46,48,37,41,47,40);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19)(18,23)(20,33)(21,37)(22,32)(24,40)(25,44)(26,45)(27,42)(28,29)(30,34)(31,39)(35,46)(36,48)(38,47)(41,43);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(18,24)(20,34)(21,38)(23,40)(25,26)(27,36)(28,43)(29,41)(30,33)(31,46)(35,39)(37,47)(42,48)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=g,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44683);