Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26,30,34)(13,19)(14,18)(15,20)(16,23)(17,29,35,28)(22,31,38,40)(24,32)(25,36)(27,33,39,37);(5,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,33)(18,36)(19,21)(20,32)(22,28)(26,40)(29,38)(30,39)(31,34)(35,37);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,34)(20,35)(21,37)(23,27)(24,39)(25,38)(32,40)(33,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22427);