Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1914 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)(3,6)(4,5)(7,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,e^a=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216083, #72216090, #72216098, … |
| maximal normal subgroup of | #72216083, #72216090, #72216098, … |
| maximal quotient of | #72240740, #72264327, #72302478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261000);