Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,8,5,3,4,7)(9,10,15,19,14,21,30,38,13,20,29,32,28,18,36,37)(11,12,23,26,22,27,35,40,24,16,31,33,39,25,17,34);(1,3,5,2)(4,7,6,8)(9,11,15,31,14,22,30,17,13,24,29,23,28,39,36,35)(10,16,32,33,21,25,37,34,20,12,19,26,18,27,38,40);(4,6)(7,8)(9,12)(10,17)(11,19)(13,16)(14,27)(15,26)(18,31)(20,35)(21,23)(22,38)(24,32)(25,28)(29,33)(30,40)(34,36)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=ei,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43991);