Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #453 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,15,26,16,35,19,11,14,27,13,10,7,17,20)(8,21,22,30,28,32,36,34,24,12,29,31,23,18,25,33);(1,2,3,4)(5,7)(6,10)(8,18)(9,15)(11,16)(12,28)(13,27)(14,26)(17,19)(20,35)(21,23)(22,30)(24,32)(25,34)(29,31)(33,36);(5,8,11,24)(6,12,14,21)(7,18,16,32)(9,25,27,36)(10,28,26,23)(13,33,15,34)(17,22,35,29)(19,30,20,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44783);