Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143719 |
| 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 | |
462 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 80 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)(11,22)(12,16)(13,20)(14,19)(15,21)(17,23)(18,26)(24,34)(25,36)(27,35)(28,32)(29,33)(30,37)(31,40)(38,39);(10,16)(11,23)(12,26)(13,14)(17,34)(18,20)(19,32)(21,33)(22,38)(24,27)(25,30)(28,29)(31,36)(35,37);(2,5)(3,8)(6,7)(9,11)(10,17)(12,27)(13,30)(14,23)(15,25)(16,31)(18,34)(19,36)(20,22)(21,37)(24,29)(26,40)(28,39)(32,35)(33,38)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,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50915);