Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150171 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | C2×#1343 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,10,7,11,24,39)(3,13,16,43,14,42,45,62)(4,9,20,33,21,35,53,37)(6,26,25,56,19,12,40,38)(8,28,30,50,29,57,58,18)(15,31,46,36,44,41,47,61)(17,49,48,64,27,32,23,55)(22,54,52,63,51,34,60,59);(1,3)(2,8)(4,18)(5,23)(6,27)(7,14)(9,32)(10,36)(11,29)(12,41)(13,34)(15,22)(16,40)(17,19)(20,47)(21,50)(24,48)(25,45)(26,31)(28,52)(30,38)(33,43)(35,49)(37,62)(39,61)(42,54)(44,51)(46,53)(55,63)(56,58)(57,60)(59,64);(3,14)(8,29)(13,42)(15,44)(16,45)(17,27)(18,50)(23,48)(28,57)(30,58)(31,41)(32,49)(36,61)(43,62)(46,47)(55,64)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^d=eh,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57391);