Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1360 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)(3,5,7,8)(4,6)(9,10)(11,20)(12,16)(13,18)(14,23)(15,19)(17,22)(21,24);(1,3)(2,5,6,8)(4,7)(9,11,14,12,15,22,13,21)(10,16,18,20,19,24,23,17);(3,7)(5,8)(9,12,14,22,15,21,13,11)(10,17,18,16,19,20,23,24);(9,13,15,14)(10,18,19,23)(11,21,22,12)(16,20,24,17)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307594, #72307612, #72307621, … |
| maximal normal subgroup of | #72307594, #72307612, #72307621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112946);