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)(10,17)(11,13)(12,22)(14,15)(18,24)(19,20);(1,3)(2,5)(4,7)(6,8)(9,10,14,17,16,20,15,19)(11,21,22,24,12,23,13,18);(9,11,14,22,16,12,15,13)(10,18,19,23,20,24,17,21);(3,7)(5,8)(9,12,15,22,16,11,14,13)(10,18,17,21,20,24,19,23);(9,13,15,12,16,22,14,11)(10,18,19,23,20,24,17,21)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=gi,d^2=g,e^2=fi,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, 7500222);