Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10464 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1440 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,14,20,15,18,13,19)(11,21,23,33,25,16,24,32)(12,26,28,35,30,36,29,17)(22,38,39,37,40,34,27,31);(1,2,6,8)(3,5,7,4)(9,11,15,25)(10,16,18,21)(12,27,30,39)(13,23,14,24)(17,34,35,38)(19,32,20,33)(22,28,40,29)(26,37,36,31);(1,3,4,2,6,7,5,8)(9,12)(10,17,18,35)(11,22)(13,28)(14,29)(15,30)(16,31,21,37)(19,26,20,36)(23,27)(24,39)(25,40)(32,34,33,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,f^c=fhi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26227);