Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 823 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,14,8,16,19,20,11,12,9,17,18,7,15,13);(2,4)(6,12)(8,18)(9,15)(10,19)(13,14)(17,20);(5,7)(6,8)(9,13)(10,20)(11,16)(12,18)(14,15)(17,19)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=g,f^2=ghi,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=eg,f^b=fghi,g^b=gi,d^c=dg,e^c=egi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219618, #72219629, #72221511, … |
| maximal normal subgroup of | #72219618, #72219629, #72224555, … |
| maximal quotient of | #72219611, #72219623, #72221511 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39032);