Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149797 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2327 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,9,15,11,17,25,34)(7,12,18,28,8,20,21,35)(10,16,26,33,27,14,24,32)(13,29,30,22,31,23,36,19);(1,2,3,4)(5,7)(6,12)(8,11)(9,18)(10,19)(13,16)(14,31)(15,28)(17,20)(21,25)(22,27)(23,24)(26,29)(30,33)(32,36)(34,35);(5,8,11,7)(6,13)(9,23,25,29)(10,19,27,22)(12,16)(14,20)(15,28)(17,31)(18,26,21,24)(30,32)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=h,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=eh,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59459);