Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,16)(11,21)(12,26)(13,14)(19,20)(22,32)(23,38)(24,25)(28,29)(30,35)(31,34)(39,40);(1,3,4,7)(2,5,6,8)(9,10,13,16,15,20,14,19)(11,22,21,17,25,32,24,33)(12,18,26,30,29,36,28,35)(23,34,37,31,38,40,27,39);(2,6)(5,8)(9,11,15,25)(10,17,20,33)(12,27,29,37)(13,24,14,21)(16,22,19,32)(18,34,36,40)(23,26,38,28)(30,39,35,31);(9,12)(10,18)(11,23)(13,26)(14,28)(15,29)(16,30)(17,31)(19,35)(20,36)(21,37)(22,34)(24,27)(25,38)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368300);