Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94193 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1391 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7237 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,29,43)(3,15,17,47,19,38,53,61)(4,12,22,44,23,10,26,39)(6,30,31,60,24,45,27,58)(8,14,33,46,32,40,28,42)(9,34,35,55,36,52,62,18)(16,49,50,41,21,37,51,63)(20,48,56,64,54,59,25,57);(1,3,6,18,32,54,23,16,7,19,24,55,8,20,4,21)(2,9,12,15,14,37,45,48,13,36,10,38,40,49,30,59)(5,25,27,51,28,53,22,34,29,56,31,50,33,17,26,52)(11,41,39,64,46,35,60,61,43,63,44,57,42,62,58,47)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=i,e^2=fghi,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1438);