Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2,3,4)(5,6)(7,8)(10,14)(15,23)(16,21)(17,24);(6,8)(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20);(13,20)(16,21)(17,24)(19,22)Polycyclic presentation:
2^9|b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72233821, #72234548, #72234569, … |
| maximal normal subgroup of | #72233821, #72234548, #72234569, … |
| maximal quotient of | #72233815, #72260903, #72280469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301359);