Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94526 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7339 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | #1820 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1820 ◁ G | |
| LECS | G ▷ #1820 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
202 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 24 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:
(1,2,5,10)(3,13,15,43,17,33,49,58)(4,18,19,51,21,52,53,63)(6,25,23,38)(7,12,24,39)(8,28,29,54)(9,32,34,55,35,62,37,20)(11,40,42,27)(14,41,45,36)(16,50,48,64,26,44,22,56)(30,60,61,47)(31,57,59,46);(1,3,7,17)(2,8,12,31)(4,14,21,47)(5,22,24,48)(6,26,27,16)(9,33,35,13)(10,36,39,61)(11,41,25,30)(15,23,49,40)(18,44,52,50)(19,28,53,57)(20,54,55,46)(29,42,59,38)(32,60,62,45)(34,56,37,64)(43,63,58,51)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=h,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=dehi,e^b=ei,f^b=fhi,d^c=dgi,f^c=fi,e^d=eh,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1723);