Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,15,18,14,11)(12,16,21,24,20,23,22,19);(1,3)(2,5)(4,8)(6,7)(9,11,14,18,15,17,13,10)(12,19,22,23,20,24,21,16);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,24)(18,23);(12,20)(16,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=be,e^a=eh,f^b=fh,d^c=di110 conjugacy classes — view the full character table.
| maximal subgroup of | #72303584 |
| maximal normal subgroup of | #72303584 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88285);