Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,16)(12,14)(13,15)(17,18,20,22)(19,21,23,24);(6,8)(9,11,13,12)(10,14,15,16);(5,6)(7,8)(17,18)(19,21)(20,22)(23,24);(1,2,3,4)(9,10)(11,14)(12,16)(13,15);(9,12,13,11)(10,14,15,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72280688 |
| maximal normal subgroup of | #72280688 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424901);