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 | |
1859 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)(11,17)(12,20)(13,21)(14,16)(15,18)(19,24)(22,23);(2,6)(7,8)(11,18)(12,21)(13,20)(15,17)(19,23)(22,24);(1,3,5,4)(2,7,6,8)(9,11,13,18,14,19,20,23)(10,15,12,17,16,22,21,24);(9,12,20,10,14,21,13,16)(11,17,23,15,19,24,18,22);(1,4,5,3)(2,7,6,8)(11,19)(15,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=gi,e^2=f,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, 7472008);