Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,6,5)(3,8,4,7)(10,16)(11,21)(12,26)(13,14)(17,31)(19,20)(23,37)(24,25)(28,29)(30,35)(32,34)(39,40);(1,3,6,4)(2,7,5,8)(9,10,13,16,15,20,14,19)(11,22,21,31,25,33,24,17)(12,18,26,30,29,36,28,35)(23,34,27,39,37,40,38,32);(1,4,6,3)(2,7,5,8)(9,11)(10,17)(12,27)(13,21)(14,24)(15,25)(16,22)(18,34)(19,33)(20,31)(23,28)(26,37)(29,38)(30,39)(32,35)(36,40);(1,5,6,2)(3,8,4,7)(9,12,15,29)(10,18,20,36)(11,23,25,37)(13,26,14,28)(16,30,19,35)(17,32,31,39)(21,27,24,38)(22,34,33,40)Polycyclic presentation:
2^9|a^2=ef,b^2=egh,c^2=ef,d^2=efi,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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380849);