Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(5,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(1,3)(2,5)(4,7)(6,8)(14,23)(17,24);(3,7)(5,8)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=di,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234697, #72234742, #72238125, … |
| maximal normal subgroup of | #72234697, #72234742, #72238125, … |
| maximal quotient of | #72234698, #72234705, #72264410 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142143);