Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,9,24)(7,13,18,12,19,28,25,27)(8,14,22,31,23,32,21,35)(17,29,33,26,20,30,34,36);(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,24)(14,26)(15,28)(16,27)(20,23)(21,34)(22,33)(29,35)(30,31)(32,36);(2,4)(5,7)(6,13)(8,20)(9,25)(10,18)(11,19)(12,15)(14,30)(16,28)(17,23)(21,33)(22,34)(24,27)(26,35)(29,32)(31,36);(5,8,10,22,11,23,9,21)(6,14,15,31,16,32,24,35)(7,17,18,33,19,20,25,34)(12,26,28,30,27,36,13,29)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72252967, #72256886, … |
| maximal normal subgroup of | #72231526, #72252967, #72256886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266264);