Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1408 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1844 | C22 |
| prime radical | C23 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1844 ◁ G | |
| LECS | G ▷ #1844 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20,22,33)(12,24,26,16)(13,18,29,17)(14,19)(15,21,30,38)(23,37,28,36)(25,32)(27,34,40,35)(31,39);(9,11,14,23)(10,15,19,30)(12,25,27,39)(13,28,29,22)(16,33,35,36)(17,32,18,31)(20,24,37,34)(21,26,38,40)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=ghi,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cgh,d^b=degi,e^b=ehi,g^b=gi,d^c=dgh,e^d=ehi,g^d=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1762);