Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| 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 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,16,13,17,12,29)(7,9,19,10,20,27,15,26)(8,14,24,21,25,33,30,32)(18,23,31,28,35,36,34,22);(5,7,9,16,12,19,26,17,13,20,27,29,11,15,10,6)(8,21,22,35,24,14,23,31,25,32,28,18,30,33,36,34);(1,2,3,4)(5,8)(6,14)(7,18)(9,23)(10,28)(11,24)(12,30)(13,25)(15,34)(16,21)(17,33)(19,31)(20,35)(22,26)(27,36)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,g^b=gi,d^c=dh,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250148, #72250154, #72250157, … |
| maximal normal subgroup of | #72250148, #72250154, #72250157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56431);