Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(9,10,15,21,16,22,14,20)(11,23,27,17,28,34,26,35)(12,29,31,38,32,37,25,18)(13,19,30,39,24,40,33,36);(1,2)(3,4)(5,7,8,6)(9,11,16,28)(10,17,22,35)(12,30,32,33)(13,25,24,31)(14,27,15,26)(18,36,38,39)(19,29,40,37)(20,34,21,23);(2,4)(6,7)(9,12,16,32)(10,18,22,38)(11,24,28,13)(14,31,15,25)(17,19,35,40)(20,29,21,37)(23,39,34,36)(26,33,27,30);(9,13,16,24)(10,19,22,40)(11,25,28,31)(12,27,32,26)(14,30,15,33)(17,29,35,37)(18,23,38,34)(20,39,21,36)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363827);