Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62800 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10)(11,18,20,24)(12,21)(13,15)(14,17)(16,19,23,22);(1,2)(3,6)(4,5)(7,8)(10,15)(16,23)(17,21)(18,24);(9,11,14,22,13,20,12,19)(10,16,17,24,15,23,21,18);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313312, #72315641, #72315677 |
| maximal normal subgroup of | #72313312, #72315641, #72315677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301080);