Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,23)(13,22)(14,19)(15,21)(17,18)(20,24);(2,6)(7,8)(9,11)(10,16)(12,24)(13,19)(14,22)(15,18)(17,21)(20,23);(1,3,5,4)(2,7,6,8)(9,12,24,11,15,22,14,18)(10,17,19,13,21,16,20,23);(9,13,24,16,15,23,14,17)(10,18,19,12,21,11,20,22);(1,4,5,3)(2,7,6,8)(9,14,15,24)(10,19,21,20)(11,22,18,12)(13,17,23,16)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472013);