Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4)(6,7)(9,10,13,17,15,20,14,19)(11,21,22,32,25,31,24,16)(12,26,27,18,29,33,28,34)(23,37,38,35,39,30,36,40);(2,4)(5,6,8,7)(9,11,15,25)(10,16,20,32)(12,23,29,39)(13,24,14,22)(17,31,19,21)(18,30,34,37)(26,40,33,35)(27,36,28,38);(2,4)(10,17)(11,22)(13,14)(18,26)(19,20)(21,31)(23,36)(24,25)(27,28)(33,34)(35,40)(38,39);(5,7,8,6)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,30)(17,26)(19,33)(20,34)(21,35)(22,36)(24,38)(25,39)(31,40)(32,37)Polycyclic presentation:
2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308276);