Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
660 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 40 points:
(1,2)(3,4,6,8)(5,7)(9,10,14,18,15,20,13,19)(11,21,23,16,25,32,24,33)(12,26,28,36,29,35,30,17)(22,34,39,40,27,31,38,37);(1,3)(2,4,7,8)(5,6)(9,11,13,24,15,25,14,23)(10,16,19,21,20,33,18,32)(12,27,28,38,29,22,30,39)(17,34,26,40,36,31,35,37);(1,4)(2,6,7,3)(5,8)(9,12,13,28,15,29,14,30)(10,17,18,35,20,36,19,26)(11,22,23,38,25,27,24,39)(16,31,21,37,33,34,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27890);