Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143938 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1999 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 458 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,23)(13,28)(14,17)(15,21)(16,26)(18,25)(20,35)(22,37)(24,34)(27,38)(29,32)(30,36)(31,39)(33,40);(1,2)(3,7)(4,6)(5,8)(15,30)(19,35)(31,34)(32,37);(2,6)(3,4)(7,8)(9,11)(10,15)(12,24)(13,29)(14,20)(16,32)(17,30)(18,34)(19,23)(21,26)(22,25)(27,39)(28,31)(33,37)(35,38)(36,40)Polycyclic presentation:
2^9|d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,d^c=dgi,e^c=eh,f^c=fg,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51134);