Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1833 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)(3,5)(4,6)(7,8)(9,10,14,20,15,19,13,22)(11,21,17,12,18,24,23,16);(1,2)(3,7)(4,6)(5,8)(9,11,14,17,15,18,13,23)(10,16,22,24,19,12,20,21);(1,3)(2,5)(4,8)(6,7)(9,12,15,16)(10,17,19,23)(11,22,18,20)(13,24,14,21);(3,8)(5,7)(9,12)(10,18)(11,19)(13,21)(14,24)(15,16)(17,22)(20,23);(2,6)(5,7)(10,19)(11,18)(17,23)(20,22)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910268);