Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(9,10)(11,18,21,24)(12,16)(13,22)(14,17)(15,20,23,19);(1,2,4,5)(3,7,8,6)(11,19)(12,13)(15,18)(16,22)(20,21)(23,24);(9,11)(10,15)(12,19)(13,20)(14,21)(16,24)(17,23)(18,22);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,19)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252481, #72252483, #72252485, … |
| maximal normal subgroup of | #72252481, #72252483, #72252485, … |
| maximal quotient of | C2×#449836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357032);