Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1677 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | 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 | |
383 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 96 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)(5,6,10,16,11,15,27,21,13,18,9,20,12,17,26,7)(8,19,23,33,29,35,28,14,24,31,36,32,22,34,25,30);(5,7,10,17,11,20,27,18,13,21,9,15,12,16,26,6)(8,19,28,32,22,35,23,30,24,31,25,33,29,34,36,14);(2,4)(5,8)(6,14)(7,19)(9,25)(10,28)(11,22)(12,29)(13,24)(15,33)(16,34)(17,32)(18,30)(20,35)(21,31)(23,27)(26,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45863);