Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93707 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1317 |
| commutator | C23×C4 | C2×C8 |
| Frattini, cosocle | #1817 | C22 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1817 ◁ G | |
| LECS | G ▷ #1817 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 30 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:
(1,2,4,11,6,13,23,38,7,12,22,40,28,41,51,63)(3,15,17,44,19,42,47,61,16,10,33,34,45,35,57,58)(5,24,25,37,26,52,53,60,8,14,21,39,27,43,50,62)(9,29,30,49,32,54,55,18,31,46,36,48,56,64,59,20);(2,9,14,15)(3,16)(4,21)(6,27)(10,12,31,24)(11,30,39,44)(13,42,43,32)(17,46)(18,20)(19,48)(22,25)(26,28)(29,33)(34,40,36,37)(35,52,56,41)(38,61,62,55)(45,49)(47,57)(54,64)(58,60,59,63)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=hi,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=dehi,f^b=fi,d^c=dg,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 903);