Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143701 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,16)(13,19)(14,28)(15,20)(18,24)(21,33)(22,29)(23,35)(25,30)(26,31)(27,34)(32,36)(37,40)(38,39);(10,16)(11,21)(12,24)(13,14)(17,32)(18,19)(20,31)(22,37)(23,27)(25,26)(28,30)(29,33)(34,40)(35,39);(2,5)(3,8)(6,7)(9,11)(10,17)(12,22)(13,27)(14,21)(15,23)(16,29)(18,32)(19,34)(20,35)(24,36)(25,38)(26,37)(28,33)(30,39)(31,40)Polycyclic presentation:
2^9|d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,e^c=eh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50897);