Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C23 | C22×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
534 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)(3,6)(4,7)(5,8)(9,10,14,20,15,17,23,11,16,21,24,22,13,19,12,18);(1,3,6,2)(4,7,5,8)(10,17,21,19)(11,20,18,22)(12,23)(14,24);(1,4,6,5)(2,7,3,8)(9,11,12,17,13,20,24,10,16,18,23,19,15,22,14,21);(1,5,6,4)(2,7,3,8)(9,12,13,24,16,23,15,14)(10,18,19,22,21,11,17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=ce,d^b=di,f^b=fi,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72313613, #72313634, #72313670, … |
| maximal normal subgroup of | #72313613, #72313634, #72313670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400525);