Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500385 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(18,21)(19,23)(22,24);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(21,24);(9,11)(10,13)(12,16)(14,15)(17,18,19,21,20,22,23,24);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(18,22)(21,24);(10,14)(13,15)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282221, #72282345, #72305295, … |
| maximal normal subgroup of | #72282221, #72282345, #72305295, … |
| maximal quotient of | #72305527, #72305536, #72305551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6414637);