Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3293 | 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 | |
794 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 794 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17,14,18,28,31,12,15,10,26,9,8,25,16)(7,19,23,36,27,34,29,32,24,13,22,33,21,20,30,35);(2,4)(5,7,12,24)(6,13,15,19)(8,20,18,34)(9,27,14,21)(10,29,11,30)(16,33,31,36)(17,32,26,35)(22,25,23,28);(5,8)(6,14)(7,20)(9,15)(10,16)(11,31)(12,18)(13,21)(17,25)(19,27)(22,35)(23,32)(24,34)(26,28)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224801, #72224806, #72224810, … |
| maximal normal subgroup of | #72224801, #72224806, #72224810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39034);