Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563341 |
| 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 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,15,18)(11,19,22,20)(12,13,17,14)(16,23,24,21);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,15,22,14,23)(10,16,12,19,18,24,17,20);(9,10,14,17,15,18,13,12)(11,20,21,16,22,19,23,24);(3,7)(5,8)(9,12,14,10,15,17,13,18)(11,16,21,19,22,24,23,20);(9,12,13,18,15,17,14,10)(11,19,21,24,22,20,23,16)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500220);