Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143216 |
| 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 | |
507 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 507 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,20)(12,26)(13,18)(14,32)(15,19)(16,22)(17,30)(21,27)(23,37)(24,35)(25,38)(28,29)(31,40)(33,39)(34,36);(2,6)(3,4)(7,8)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,35)(18,21)(19,33)(20,26)(22,30)(28,38)(29,39)(31,36)(32,34)(37,40);(2,7)(3,4)(6,8)(9,10)(11,21)(12,28)(13,32)(14,18)(15,19)(16,25)(17,31)(20,36)(22,35)(23,34)(24,33)(26,29)(27,37)(30,40)(38,39)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50412);