Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,15,20,13,17,14,19)(11,16,24,32,22,29,23,31)(12,25,28,18,26,34,27,33)(21,35,38,30,36,39,37,40);(9,11,15,23,13,22,14,24)(10,16,19,32,17,29,20,31)(12,21,27,38,26,36,28,37)(18,30,34,35,33,40,25,39);(1,2,3,4)(10,17)(16,29)(18,33)(19,20)(25,34)(30,40)(31,32)(35,39);(6,8)(9,12)(10,18)(11,21)(13,26)(14,28)(15,27)(16,30)(17,33)(19,34)(20,25)(22,36)(23,38)(24,37)(29,40)(31,39)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,g^b=gi,h^b=hi,e^d=ei,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300264);