Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #33713 | 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 | |
450 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,5)(4,7)(6,8)(9,10,15,22,16,23,13,20,17,24,33,39,25,34,14,21)(11,19,27,18,28,36,12,30,29,35,32,37,31,40,26,38);(9,11,25,28,17,29,16,31)(10,18,34,30,24,37,23,38)(12,14,32,33,26,13,27,15)(19,20,36,22,35,21,40,39);(2,5)(3,8)(6,7)(9,12)(10,19)(11,14)(13,29)(15,31)(16,27)(17,26)(18,20)(21,37)(22,30)(23,40)(24,35)(25,32)(28,33)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=ghi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44731);