Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125636 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 56 points:
(5,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,30,34)(27,35,38,54)(28,39,43,53)(29,44,41,33)(31,46,48,36)(32,49,51,42)(37,55,45,50)(40,52,56,47);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18);(1,3,4,7)(2,5,6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,27,30,38)(26,31,34,48)(28,40,43,56)(29,45,41,37)(32,50,51,55)(33,52,44,47)(35,53,54,39)(36,49,46,42)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=eg,d^c=dg,e^c=eh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33195);