Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,16,12,21)(11,18,19,15,20,24,22,23);(2,5)(6,8);(9,11)(10,15)(12,22)(13,19)(14,20)(16,23)(17,24)(18,21);(1,3,4,7)(2,6,5,8);(10,16)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|a^2=h,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224743, #72224748, #72224749, … |
| maximal normal subgroup of | #72224743, #72224748, #72224749, … |
| maximal quotient of | #72226691, #72239678, #72239725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278193);