Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131844 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8443 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,11,5,9,20,10,6,12,21,28,22,7,18,25)(3,13,16,27,23,31,15,26,17,30,19,24,14,29,32,8);(2,7)(4,18)(5,22)(8,24)(9,12)(11,28)(13,30)(14,23)(15,19)(16,32)(20,21)(26,27);(1,3)(2,8)(4,19)(5,23)(6,17)(7,24)(9,27)(10,29)(11,30)(12,26)(13,28)(14,22)(15,18)(16,21)(20,32)(25,31)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=g,f^2=gh,g^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=egi,f^b=fghi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286402 |
| maximal normal subgroup of | #72286402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39040);