Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137659 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1038 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)(5,6,9,16,11,15,10,28)(7,17,19,34,20,13,27,33)(8,21,24,35,26,32,25,14)(12,22,30,23,29,36,31,18);(5,7)(6,12)(8,22)(9,19)(10,27)(11,20)(13,32)(14,33)(15,29)(16,30)(17,21)(18,25)(23,24)(26,36)(28,31)(34,35);(2,4)(5,7)(6,13)(8,23)(9,27)(10,19)(11,20)(12,21)(14,30)(15,17)(16,34)(18,26)(22,24)(25,36)(28,33)(29,32)(31,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dh,h^b=hi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231484, #72231549, #72231553, … |
| maximal normal subgroup of | #72231484, #72231549, #72231553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54834);