Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150723 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11839 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | #534 | C2×C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 393 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 — 3 generators on 64 points:
(1,2,6,11,7,12,27,40)(3,13,16,42,17,35,50,59)(4,18,20,39,21,9,26,33)(5,22,24,53,25,10,34,37)(8,28,30,15,31,47,38,48)(14,43,45,63,46,41,49,62)(19,32,52,60,23,36,55,61)(29,56,57,54,51,64,58,44);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,29)(10,35)(11,38)(12,31)(13,22)(18,51)(19,44)(20,45)(21,46)(23,54)(24,28)(25,48)(26,49)(27,50)(30,40)(32,43)(33,58)(34,47)(36,41)(37,59)(39,57)(42,53)(52,64)(55,56)(60,63)(61,62);(1,4,7,21)(2,9,12,18)(3,14,17,46)(5,23,25,19)(6,26,27,20)(8,29,31,51)(10,36,22,32)(11,39,40,33)(13,41,35,43)(15,44,48,54)(16,49,50,45)(24,52,34,55)(28,56,47,64)(30,58,38,57)(37,60,53,61)(42,63,59,62)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,h^b=hi,d^c=dhi,e^c=ei,e^d=eh,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57922);