Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C23 | #440 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(10,16)(11,21)(12,13)(14,26)(17,33)(18,19)(20,32)(22,23)(24,40)(25,29)(27,28)(30,39)(31,35)(36,37);(1,3,5,6,4,7,8,2)(9,10)(11,17)(12,25)(13,16)(14,29)(15,20)(18,27)(19,26)(21,34)(22,31)(23,33)(24,37)(28,32)(30,36)(35,39)(38,40);(2,6)(3,7)(9,11,15,24)(10,17,20,37)(12,21,27,40)(13,22,28,39)(14,30,26,23)(16,31,32,35)(18,38,25,34)(19,33,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ch,d^b=dg,e^b=eg,g^b=gi,d^c=dg,e^c=egi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43120);