Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149789 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,18,31)(8,19,21,35)(9,23,25,13)(10,14)(11,17,27,33)(12,22)(16,30,24,32)(20,28,34,26)(29,36);(1,2,3,4)(7,16)(11,26)(15,30)(17,20)(18,24)(27,28)(31,32)(33,34);(5,7)(6,11,14,26)(8,20)(9,24)(10,16)(12,15,29,30)(13,27,23,28)(17,21)(18,25)(19,31,35,32)(22,34)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=ehi,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59442);