Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(2,4)(5,6)(7,15,16,33)(8,20,22,21)(9,13,24,31)(10,14)(11,17,25,32)(12,28)(18,34,19,29)(23,30)(26,35,27,36);(1,2,3,4)(5,6)(7,11)(8,21)(9,13)(10,14)(12,23)(15,32)(16,25)(17,33)(18,26)(19,27)(20,22)(24,31)(28,30)(29,35)(34,36);(5,7,10,19)(6,11,14,27)(8,17,22,36)(9,16,24,18)(12,29,30,15)(13,25,31,26)(20,34,21,33)(23,35,28,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=ehi,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59772);