Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188759 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,17,19,29,20,12,24,27)(8,21,23,28,22,30,25,13)(18,32,35,31,33,36,34,26);(1,2)(3,4)(5,7,9,19,11,20,10,24)(6,12,15,27,14,17,16,29)(8,18,23,34,22,33,25,35)(13,26,21,36,28,31,30,32);(2,4)(5,8)(6,13,14,28)(7,18)(9,23)(10,25)(11,22)(12,26,17,31)(15,21,16,30)(19,34)(20,33)(24,35)(27,36,29,32);(8,22)(13,28)(18,33)(21,30)(23,25)(26,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252118, #72252123, #72252136, … |
| maximal normal subgroup of | #72252118, #72252123, #72252136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265466);