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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,24,20,23);(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,20)(18,23,21,24);(1,2,3,4)(5,6,7,8)(17,19,22,20)(18,23,21,24);(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,23,21,24);(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72327952, #72327961, #72327964, … |
| maximal normal subgroup of | #72327952, #72327961, #72327964, … |
| maximal quotient of | #72335935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 425033);