Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,20)(11,21,25,32)(12,26,30,36)(13,18,14,19)(16,23,33,24)(17,29,35,28)(22,31,39,40)(27,34,38,37);(1,3,5,8)(2,6,4,7)(9,11,15,25)(10,16,20,33)(12,27,30,38)(13,23,14,24)(17,34,35,37)(18,21,19,32)(22,28,39,29)(26,31,36,40);(1,2,5,4)(3,6,8,7)(9,12,15,30)(10,17,20,35)(11,22,25,39)(13,29,14,28)(16,31,33,40)(18,26,19,36)(21,37,32,34)(23,38,24,27);(1,4,5,2)(3,6,8,7)(10,18)(11,23)(13,14)(17,26)(19,20)(21,32)(22,38)(24,25)(27,39)(28,29)(34,37)(35,36)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,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,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339824);