Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631112 |
| 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 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(18,21)(19,22)(23,24);(3,7)(5,8)(9,11,13,12)(10,14,16,15)(17,18,20,21)(19,23,24,22);(1,3)(2,5)(4,7)(6,8)(10,15)(11,12)(14,16)(17,19)(18,22)(20,24)(21,23);(2,6)(3,7)(9,11,13,12)(10,15,16,14)(17,18,20,21)(19,22,24,23);(10,16)(14,15)(17,18,20,21)(19,22,24,23)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^c=dfh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538578);