Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184991 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3494 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,13,16,14,17,12,18)(11,19,20,23,21,15,22,24);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(2,5)(6,8)(10,16)(12,13)(15,23)(17,18)(19,24)(20,22);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252477, #72252480, #72252485, … |
| maximal normal subgroup of | #72252477, #72252480, #72252485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271089);