Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1786 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)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(5,6)(7,8)(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(2,4)(6,8);(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249925, #72249942, #72249947, … |
| maximal normal subgroup of | #72249925, #72249942, #72249947, … |
| maximal quotient of | #72264821, C2×#498795, #72322502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405991);