Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #33579 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 384 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,7,5,3,4,8)(9,10,15,22,16,23,14,21,17,24,34,40,33,39,13,20)(11,25,27,37,28,38,12,30,29,19,32,18,31,35,26,36);(1,3,5,2)(4,8,6,7)(9,11,16,31,17,29,33,28)(10,18,23,30,24,37,39,36)(12,14,27,34,26,13,32,15)(19,20,38,22,25,21,35,40);(1,4,5,6)(2,7,3,8)(9,12,17,26)(10,19,24,25)(11,14,29,13)(15,31,34,28)(16,27,33,32)(18,20,37,21)(22,36,40,30)(23,38,39,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=gh,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44121);