Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,18,11,17,10,16)(7,19,22,36,21,35,20,34)(8,13,25,32,24,31,23,14)(9,26,29,33,28,15,27,30);(1,2)(3,4)(5,7,9,8,11,21,28,24)(6,13,15,19,17,31,26,35)(10,20,27,23,12,22,29,25)(14,33,34,18,32,30,36,16);(2,4)(5,8,9,21,11,24,28,7)(6,14,15,34,17,32,26,36)(10,25,27,20,12,23,29,22)(13,30,19,16,31,33,35,18)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=ei,g^c=gh,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55898);