Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150413 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | #11899 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,6,12,7,14,30,44)(3,15,18,46,20,47,53,64)(4,21,22,55,24,42,28,59)(5,10,26,39,23,11,40,41)(8,13,29,45,31,38,57,63)(9,32,34,50,35,56,62,25)(16,36,49,61,17,51,52,43)(19,33,54,60,48,37,27,58);(1,3,7,20)(2,9,14,35)(4,16)(5,25)(6,27)(8,19,31,48)(10,37)(11,33)(12,43)(13,36,38,51)(15,21)(17,24)(18,29)(22,56,28,32)(23,50)(26,49,40,52)(30,54)(34,45)(39,46,41,64)(42,47)(44,61)(53,57)(55,60,59,58)(62,63);(1,4,8,23,7,24,31,5)(2,10,38,42,14,11,13,21)(3,16,48,25,20,17,19,50)(6,28,29,26,30,22,57,40)(9,33,36,47,35,37,51,15)(12,41,63,55,44,39,45,59)(18,52,27,56,53,49,54,32)(34,58,61,46,62,60,43,64)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gh,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfi,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58099);