Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33779 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
523 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 70 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)(10,17)(11,23)(12,28)(13,31)(14,15)(18,34)(19,20)(21,29)(22,32)(24,25)(26,39)(27,30)(33,36)(37,38);(2,5)(3,8)(6,7)(9,10,16,22)(11,18,27,38)(12,29,31,19)(13,32,28,17)(14,21,15,20)(23,35,30,40)(24,36,39,33)(25,34,26,37);(3,8)(5,6)(9,11)(10,18)(12,30)(13,24)(14,26)(15,25)(16,27)(17,33)(19,35)(20,34)(21,37)(22,38)(23,31)(28,39)(29,40)(32,36)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51282);