Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36);(1,2,3,4)(6,9)(7,11)(13,15)(14,17)(16,19)(18,20)(23,29)(24,31)(26,32)(27,34)(30,35)(33,36);(7,11)(10,12)(14,18)(17,20)(21,23,24,29,25,30,31,35)(22,26,27,32,28,33,34,36);(5,7,8,11)(6,10,9,12)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=efi,b^2=h,c^2=g,d^2=e,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gi,d^c=de80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299283);