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 | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1133 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)(11,19,22,33)(12,16)(13,17)(14,18)(15,21,29,25)(20,36,38,40)(23,31)(24,32)(26,34)(27,35)(28,39,30,37);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,30)(17,33)(18,29)(19,35)(21,27)(23,39)(24,38)(26,37)(28,32)(31,40)(34,36);(1,4)(2,3)(5,8)(6,7)(9,12,14,24)(10,16,18,32)(11,20,22,38)(13,26,27,23)(15,28,29,30)(17,34,35,31)(19,36,33,40)(21,39,25,37)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^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, 19178);