Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,13,11,16,9,15)(7,17,20,28,21,26,19,12)(8,22,24,30,25,29,23,14)(18,32,34,36,35,27,33,31);(1,2,4,3)(5,7)(6,12,16,28)(8,18)(9,19)(10,20)(11,21)(13,17,15,26)(14,27,30,32)(22,31,29,36)(23,33)(24,34)(25,35);(6,13,16,15)(12,26,28,17)(14,22,30,29)(27,36,32,31);(1,3,4,2)(5,8,9,23,11,25,10,24)(6,14,13,22,16,30,15,29)(7,18,19,33,21,35,20,34)(12,27,17,31,28,32,26,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280138, #72280309, #72280417 |
| maximal normal subgroup of | #72280138, #72280309, #72280417 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275615);