Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143450 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33719 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(2,5)(3,7)(6,8)(9,10)(11,22)(12,14)(13,29)(15,21)(16,20)(17,24)(18,32)(19,33)(23,26)(25,38)(27,34)(28,40)(30,36)(31,37)(35,39);(3,7)(5,8)(9,11)(10,17)(12,28)(13,30)(14,27)(15,26)(16,25)(18,23)(19,38)(20,36)(21,35)(22,29)(24,32)(31,39)(33,40)(34,37);(1,2)(3,8)(4,6)(5,7)(9,12,14,10)(11,23,27,38)(13,29,31,37)(15,20,16,21)(17,30,28,39)(18,33,19,32)(22,26,34,25)(24,36,40,35)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50647);