Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| 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 | |
2026 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)(11,20)(12,16)(13,17)(14,19)(15,22)(18,24)(21,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,14)(13,22)(16,18)(17,20)(19,21)(23,24);(9,12,14,22,13,23,24,11)(10,15,18,16,17,20,19,21);(1,4,5,3)(2,6,7,8)(9,12,13,23)(10,16,17,21)(11,14,22,24)(15,18,20,19);(9,10)(11,21)(12,15)(13,17)(14,18)(16,22)(19,24)(20,23)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=f,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=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491523);