Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2044 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)(3,4)(5,6)(7,8)(9,10,13,17,14,18,12,15)(11,16,21,23,19,24,20,22);(2,4)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(6,8);(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231444, #72252967, #72252972, … |
| maximal normal subgroup of | #72231444, #72252967, #72252972, … |
| maximal quotient of | #72286793, C2×#354302, #72318216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261498);