Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143942 |
| 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 | |
439 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 439 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,22)(12,21)(13,28)(14,20)(15,18)(16,23)(17,24)(19,32)(25,37)(26,35)(27,33)(29,34)(30,39)(31,38)(36,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,29)(14,25)(15,23)(17,19)(18,37)(20,35)(21,22)(24,31)(27,30)(28,36)(32,40)(33,34)(38,39);(2,6)(3,4)(7,8)(9,11)(10,17)(12,23)(13,30)(14,25)(15,26)(16,33)(18,36)(19,35)(20,34)(21,39)(22,28)(24,32)(27,29)(31,40)(37,38)Polycyclic presentation:
2^9|d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dgi,e^c=eh,f^c=fgi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51138);