Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,15,19,14,20)(11,21,23,37,25,38,24,34)(12,26,28,17,30,35,29,36)(16,27,31,40,32,39,33,22);(2,6,8,7)(9,11)(10,16,19,32)(12,27)(13,23)(14,24)(15,25)(17,34,36,37)(18,31,20,33)(21,26,38,35)(22,29)(28,40)(30,39);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,d^b=dh,g^b=gi,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34745);