Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2099 |
| 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 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17,12,19,30,18)(7,20,25,36,8,26,24,31)(9,16,10,15,21,34,27,35)(13,29,32,22,33,23,14,28);(1,2)(3,4)(5,7,10,23,30,25,9,28,12,8,27,29,11,24,21,22)(6,13,16,31,17,14,15,26,19,33,34,36,18,32,35,20);(2,4)(5,8,9,28,11,25,10,23,12,7,21,22,30,24,27,29)(6,14,15,31,18,13,16,20,19,32,35,36,17,33,34,26)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250565, #72250580, #72250598, … |
| maximal normal subgroup of | #72250565, #72250580, #72250598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56187);