Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| 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)(7,21)(8,25)(9,19)(10,18)(11,33)(12,16)(13,20)(14,22)(15,27)(17,32)(23,28)(24,26)(29,36)(30,31)(34,35);(5,7,9,22,13,24,12,25)(6,14,16,21,20,8,19,26)(10,30,11,34,29,23,32,15)(17,31,36,27,33,28,18,35);(5,8,9,26,13,14,12,21)(6,15,16,30,20,34,19,23)(7,18,22,17,24,36,25,33)(10,31,11,27,29,28,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250071, #72250080, #72250104, … |
| maximal normal subgroup of | #72250071, #72250080, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55969);