Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150413 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | #11873 |
| prime radical | #24287 | C2 |
| 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 | |
411 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 411 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:
(1,2,6,11)(3,14,17,46)(4,20,21,53)(5,10,25,38)(7,30,29,59)(8,12,28,43)(9,32,34,23)(13,44,42,24)(15,45,48,62)(16,36,50,40)(18,33,52,60)(19,47,51,63)(22,54,56,64)(26,39,57,41)(27,58,31,37)(35,55,61,49);(1,3)(2,9)(4,15)(5,23)(6,27)(7,31)(8,19)(10,37)(11,40)(12,35)(13,45)(14,20)(16,22)(17,44)(18,24)(21,55)(25,48)(26,49)(28,52)(29,51)(30,36)(32,56)(33,39)(34,42)(38,46)(41,63)(43,62)(47,54)(50,57)(53,60)(58,64)(59,61);(1,4,7,5,8,22,24,26)(2,10,13,20,12,39,30,54)(3,15,18,49,19,16,31,23)(6,21,29,25,28,56,44,57)(9,33,36,14,35,37,45,47)(11,38,42,53,43,41,59,64)(17,48,52,55,51,50,27,32)(34,60,40,46,61,58,62,63)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dh,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, 57997);