Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4⋊#95 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,8,5,7,6,3)(9,10)(11,20)(12,25)(13,29)(14,18)(15,19)(16,22)(17,26)(21,30)(23,32)(24,31)(27,33)(28,39)(34,36)(35,38)(37,40);(1,3,6,7,5,8,4,2)(9,11,14,22,15,24,13,23)(10,16,18,31,19,32,29,20)(12,21,27,35,28,37,26,36)(17,30,25,38,33,40,39,34);(1,4,5,6)(2,3,7,8)(9,12,13,26,15,28,14,27)(10,17,18,25,19,33,29,39)(11,21,23,36,24,37,22,35)(16,30,31,38,32,40,20,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16371);