Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,15,22,17,23,16,35)(11,24,27,40,28,19,34,39)(12,29,26,18,30,37,36,38)(13,31,32,21,14,33,25,20);(1,2)(3,8)(4,6)(5,7)(9,11,15,27,17,28,16,34)(10,18,22,37,23,38,35,29)(12,25,26,13,30,32,36,14)(19,20,39,31,24,21,40,33);(3,7)(5,8)(9,12)(10,19)(11,25)(13,27)(14,34)(15,26)(16,36)(17,30)(18,20)(21,38)(22,39)(23,24)(28,32)(29,33)(31,37)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43696);