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 | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 256 |
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)(25,26,30,35)(27,31,38,46)(28,32,41,48)(29,42,40,34)(33,49,47,39)(36,43,52,55)(37,53,51,45)(44,56,54,50);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,27,30,38)(26,31,35,46)(28,36,41,52)(29,37,40,51)(32,43,48,55)(33,44,47,54)(34,45,42,53)(39,50,49,56);(1,3)(2,5)(4,7)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(26,32)(28,39)(29,40)(31,43)(33,34)(35,48)(36,50)(37,51)(41,49)(42,47)(44,45)(46,55)(52,56)(53,54)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,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, 33097);