Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle | C24 | C23×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,15)(12,14)(13,16)(17,18,20,22,19,21,23,24);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,21)(22,24);(2,4)(11,16)(13,15);(10,14)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72310247, #72310248, #72310255, … |
| maximal normal subgroup of | #72310247, #72310248, #72310255, … |
| maximal quotient of | #72317183 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262252);