Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2093 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,11,15,10,25)(7,16,21,31,20,32,18,34)(8,22,24,30,23,36,17,13)(12,26,29,35,28,33,27,19);(1,2)(3,4)(5,7,17,33,10,18,23,35,11,20,24,26,9,21,8,19)(6,12,22,31,14,29,30,32,15,28,36,34,25,27,13,16);(2,4)(7,18,20,21)(8,23)(12,27,28,29)(13,30)(16,31,32,34)(17,24)(19,26,35,33)(22,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250158, #72250163, #72250177, … |
| maximal normal subgroup of | #72250158, #72250163, #72250177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56403);