Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1808 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10,14,18,15,12,26,19)(11,20,24,36,25,38,40,37)(13,28,29,17,30,32,21,27)(16,31,33,35,22,39,34,23);(9,11,21,39,14,24,13,23,15,25,29,31,26,40,30,35)(10,16,32,36,19,34,17,20,12,22,28,37,18,33,27,38);(9,12)(10,15)(11,22)(13,27)(14,18)(16,25)(17,30)(19,26)(20,35)(21,28)(23,38)(24,33)(29,32)(31,36)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=dgi,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41874);