Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1442 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,3,5,7,8,6)(9,10)(11,20)(12,24)(13,29)(14,18)(15,19)(16,22)(17,27)(21,34)(23,32)(25,39)(26,35)(28,38)(30,33)(31,40)(36,37);(2,3,7,6)(9,11)(10,16)(12,25,27,40)(13,30)(14,22)(15,23)(17,34,24,36)(18,32)(19,33)(20,29)(21,28,37,26)(31,35,39,38);(1,3,4,2,5,6,8,7)(9,12,13,26,15,27,14,28)(10,17,18,35,19,24,29,38)(11,21,22,25,23,37,30,40)(16,31,20,36,33,39,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^c=eh,f^c=fhi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28091);