Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478213 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 793 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,20,15,19,28,21)(11,16,25,34,24,35,26,36)(12,27,29,17,30,18,13,31)(22,37,39,32,38,33,23,40);(1,3,5,4)(2,6,7,8)(9,11,14,26,15,24,28,25)(10,16,21,34,19,35,20,36)(12,23,29,38,30,39,13,22)(17,33,27,40,31,37,18,32);(1,4,5,3)(2,6,7,8)(9,12,28,29,15,30,14,13)(10,17,20,27,19,31,21,18)(11,22,25,23,24,38,26,39)(16,32,36,33,35,40,34,37);(3,4)(6,8)(9,13,14,30,15,29,28,12)(10,18,20,17,19,27,21,31)(11,23,25,38,24,39,26,22)(16,33,34,32,35,37,36,40)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=efi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387029);