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)(11,20,24,35,25,37,40,36)(13,28,21,17,29,33,38,27)(16,30,31,34,22,39,32,23);(9,11,21,30,14,24,29,34,15,25,38,39,26,40,13,23)(10,16,28,36,19,32,27,37,12,22,33,35,18,31,17,20);(9,12)(10,15)(11,22)(13,27)(14,18)(16,25)(17,29)(19,26)(20,34)(21,33)(23,37)(24,31)(28,38)(30,35)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=dg,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, 41873);