Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,19)(11,20,24,32)(12,25,28,36)(13,16,29,18)(15,23,33,22)(17,30,35,27)(21,38,39,31)(26,34,40,37);(1,3,5,8)(2,6,4,7)(9,11,14,24)(10,15,19,33)(12,26,28,40)(13,23,29,22)(16,20,18,32)(17,34,35,37)(21,30,39,27)(25,31,36,38);(1,2,5,4)(3,6,8,7)(10,16)(13,29)(15,20)(17,25)(18,19)(22,23)(26,40)(27,30)(31,37)(32,33)(34,38)(35,36);(1,4,5,2)(3,6,8,7)(9,12,14,28)(10,17,19,35)(11,21,24,39)(13,30,29,27)(15,31,33,38)(16,25,18,36)(20,37,32,34)(22,40,23,26)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=efi,d^2=ef,e^2=i,f^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,e^c=ei,f^c=fi,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, 339820);