Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111981 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1159 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)(3,4,7,8)(5,6)(9,10,14,20,15,19,13,18)(11,21,24,16,25,31,23,30)(12,17,26,33,27,34,28,35)(22,36,37,29,38,40,39,32);(1,3)(2,4)(5,7)(6,8)(9,11,15,25)(10,16,19,30)(12,22,27,38)(13,24,14,23)(17,32,34,29)(18,31,20,21)(26,39,28,37)(33,40,35,36);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,35)(19,34)(20,33)(21,36)(23,39)(24,37)(25,38)(30,32)(31,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19182);