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 | #1786 |
| prime radical | #61433 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 589 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,20,16,21,15,34)(11,22,24,38,12,25,27,37)(13,28,31,39,33,40,32,19)(17,29,35,26,18,30,36,23);(1,2)(3,7)(4,6)(5,8)(9,11,16,12)(10,17,21,18)(13,29,33,30)(14,24,15,27)(19,37,39,38)(20,35,34,36)(22,40,25,28)(23,31,26,32);(9,12,16,11)(10,18,21,17)(13,30,33,29)(14,27,15,24)(19,38,39,37)(20,36,34,35)(22,28,25,40)(23,32,26,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=eg,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, 38987);