Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,7,8)(9,10)(11,21)(12,25)(13,16)(14,19)(15,20)(17,22)(18,26)(23,31)(24,30)(27,34)(28,33)(29,32)(35,37)(36,39)(38,40);(1,2,3,4)(10,16)(11,22)(13,14)(18,25)(19,20)(21,30)(23,36)(24,29)(26,28)(33,34)(35,39)(37,38);(2,4)(6,8)(9,11)(10,17)(12,23)(13,22)(14,29)(15,24)(16,30)(18,31)(19,21)(20,32)(25,39)(26,37)(27,38)(28,36)(33,40)(34,35);(9,12,15,27)(10,18,20,33)(11,23,24,38)(13,28,14,26)(16,25,19,34)(17,31,32,40)(21,35,30,39)(22,36,29,37)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273938);