Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #354 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7205 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,4,10,6,9,23,34,7,13,20,38,28,12,22,41)(3,14,15,36,17,33,19,51,18,32,44,60,50,8,29,30)(5,24,25,37,26,40,43,61,27,11,21,39,53,35,54,57)(16,46,47,62,48,31,56,59,49,42,45,63,64,55,52,58);(1,3,5,16,6,17,26,48,7,18,27,49,28,50,53,64)(2,8,11,31,9,14,35,42,13,33,24,55,12,32,40,46)(4,19,21,52,23,44,54,47,20,29,25,56,22,15,43,45)(10,36,37,62,34,51,61,59,38,60,39,63,41,30,57,58)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dfhi,d^c=dhi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1357);