Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94622 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7394 |
| commutator | #381 | C2×C4 |
| Frattini, cosocle | #1782 | C22 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ #381 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1782 ◁ G | |
| LECS | G ▷ #1782 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 22 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:
(1,2,5,10,7,12,24,38)(3,13,15,41,17,33,47,57)(4,18,19,49,21,50,52,63)(6,11,23,37,27,40,46,62)(8,28,29,44,31,51,59,48)(9,32,34,26,35,55,61,20)(14,39,43,36,45,30,60,58)(16,42,22,53,25,56,54,64);(1,3,7,17)(2,8,12,31)(4,14,21,45)(5,22,24,54)(6,25,27,16)(9,33,35,13)(10,36,38,58)(11,39,40,30)(15,46,47,23)(18,42,50,56)(19,51,52,28)(20,48,26,44)(29,37,59,62)(32,43,55,60)(34,53,61,64)(41,63,57,49)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=f,d^2=i,e^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defi,e^b=ei,g^b=gh,d^c=dgi,e^c=ehi,g^c=gi,f^d=fi26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1821);