Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3738 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(1,3,4,8)(2,5,6,7);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224728, #72224735, #72302898, … |
| maximal normal subgroup of | #72224728, #72224735, #72302898, … |
| maximal quotient of | #72303898 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95276);