Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
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 | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 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)(6,8)(9,10,14,18,15,19,13,16)(11,20,12,17,21,23,22,24);(2,4)(10,16,19,18)(11,21)(12,22)(17,23,24,20);(5,6,7,8)(9,11,14,12,15,21,13,22)(10,17,16,20,19,24,18,23);(9,12,14,21,15,22,13,11)(10,17,18,23,19,24,16,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226894, #72226931, #72237283, … |
| maximal normal subgroup of | #72226894, #72226931, #72237283, … |
| maximal quotient of | #72237281, #72279972, C2×#354640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261836);