Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1586 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)(7,16,19,12)(8,21)(9,26)(10,15)(11,14)(13,24)(17,29,34,36)(18,28,20,33)(22,27,35,32)(23,30)(25,31);(2,4)(5,7,10,20,11,19,9,18)(6,12,15,28,14,16,26,33)(8,22,24,17,23,35,25,34)(13,29,30,32,31,36,21,27);(5,8,9,24,11,23,10,25)(6,13,15,21,14,31,26,30)(7,17,18,35,19,34,20,22)(12,27,28,36,16,32,33,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,g^b=gi,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72255615, #72255623, #72255635, … |
| maximal normal subgroup of | #72255615, #72255623, #72255635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31188);