Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 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)(7,16,19,12)(8,21)(9,25)(10,15)(11,14)(13,23)(17,31,32,35)(18,27,20,30)(22,36)(24,28)(26,33,29,34);(5,7,10,20,11,19,9,18)(6,12,15,27,14,16,25,30)(8,17,24,34,22,32,23,33)(13,26,21,35,28,29,36,31);(2,4)(5,8)(6,13,14,28)(7,17)(9,23)(10,24)(11,22)(12,26,16,29)(15,21,25,36)(18,33)(19,32)(20,34)(27,35,30,31);(8,22)(13,28)(17,32)(21,36)(23,24)(26,29)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252153, #72252160, #72252164, … |
| maximal normal subgroup of | #72252153, #72252160, #72252164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260740);