Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111959 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35076 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,16,14,17,12,21)(11,18,19,15,20,24,22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21);(1,2)(3,5)(4,6)(7,8)(15,23)(18,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226724, #72232727, #72234083, … |
| maximal normal subgroup of | #72226724, #72232727, #72234083, … |
| maximal quotient of | #72232725, #72233588, #72239975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19159);