Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1525 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)(9,10,14,20,16,21,15,19)(11,22,24,17,13,31,25,34)(12,26,28,38,30,37,29,18)(23,39,27,35,32,33,40,36);(1,2)(3,4)(9,11,14,24,16,13,15,25)(10,17,19,22,21,34,20,31)(12,27,28,32,30,40,29,23)(18,35,37,39,38,36,26,33);(2,4)(9,12,16,30)(10,18,21,38)(11,23,13,32)(14,29,15,28)(17,33,34,39)(19,26,20,37)(22,36,31,35)(24,40,25,27);(6,8)(9,13,15,24,16,11,14,25)(10,17,20,31,21,34,19,22)(12,27,29,23,30,40,28,32)(18,36,26,39,38,35,37,33)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362308);