Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2725 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,16)(13,17)(14,15)(19,23)(20,21)(22,24);(1,3,5,4)(2,6,7,8)(9,11,13,15)(10,14,17,18)(12,20,21,16)(19,22,24,23);(10,15)(11,17)(14,18)(16,24)(19,20)(22,23);(1,4,5,3)(2,6,7,8)(9,10)(11,14)(12,19)(13,17)(15,18)(16,23)(20,22)(21,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,24)(17,20)(18,23)Polycyclic presentation:
2^9|b^2=f,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483928);