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 | |
203 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 203 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)(3,14,16,46)(4,10,21,38)(6,28,29,60)(7,13,27,42)(8,31,32,43)(9,33,34,53)(12,26,44,30)(15,45,48,40)(17,37,51,62)(18,35,50,61)(19,54,55,64)(20,36,56,63)(22,39,49,41)(23,57,25,59)(24,58,52,47);(1,3,6,17,30,52,22,15,7,18,23,53,8,19,4,20)(2,9,10,35,31,45,57,47,13,37,39,14,12,36,28,54)(5,24,25,56,26,50,21,51,27,55,29,48,32,16,49,33)(11,40,41,64,43,62,60,61,42,63,38,58,44,34,59,46)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,e^2=fgh,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, 1436);