Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #3100 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,3,7,5,8,4,6)(9,10,13,19,15,20,30,28,14,18,21,35,31,11,16,23)(12,24,26,36,27,22,32,37,25,17,29,33,40,34,39,38);(2,6)(3,4)(7,8)(9,11,21,23,15,10,16,19,14,20,13,28,31,18,30,35)(12,22,26,38,27,17,32,36,25,34,29,37,40,24,39,33);(2,7)(3,4)(6,8)(9,12)(10,17)(11,22)(13,29)(14,25)(15,27)(16,32)(18,24)(19,36)(20,34)(21,26)(23,38)(28,37)(30,39)(31,40)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egi,f^c=fg86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43870);