Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137493 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 446 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)(9,10,15,22,35,36,34,20,17,24,14,21,16,23,13,33)(11,25,28,18,32,19,26,39,30,40,27,38,29,37,12,31);(9,11,16,32,17,30,35,29)(10,18,36,31,24,38,23,39)(12,13,28,14,26,34,27,15)(19,22,25,20,37,21,40,33);(2,5)(3,8)(6,7)(9,12)(10,19)(11,13)(14,32)(15,29)(16,28)(17,26)(18,22)(20,31)(21,38)(23,40)(24,37)(25,36)(27,35)(30,34)(33,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=ghi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44740);