Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143219 |
| 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 |
| prime radical | #19187 | C2 |
| 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 | |
527 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,7)(5,8)(10,16)(12,25)(13,14)(17,31)(18,26)(19,32)(20,33)(21,37)(22,38)(23,30)(27,39)(28,29)(34,35)(36,40);(2,6)(3,4)(7,8)(9,10,15,20)(11,21,24,36)(12,26,29,32)(13,19,14,18)(16,28,33,25)(17,30,35,23)(22,31,27,34)(37,39,40,38);(2,7)(3,4)(6,8)(9,11)(10,17)(12,27)(13,30)(14,23)(15,24)(16,31)(18,36)(19,21)(20,35)(22,29)(25,39)(26,40)(28,38)(32,37)(33,34)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50416);