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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1727 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)(9,10,13,19,15,20,14,16)(11,21,24,33,25,32,22,17)(12,26,27,18,29,30,28,34)(23,37,36,35,39,31,38,40);(2,4)(5,6,8,7)(10,16)(11,22)(13,14)(18,30)(19,20)(21,32)(23,36)(24,25)(26,34)(27,28)(35,40)(38,39);(2,4)(9,11,15,25)(10,17,20,33)(12,23,29,39)(13,22,14,24)(16,21,19,32)(18,31,34,37)(26,40,30,35)(27,38,28,36);(5,7,8,6)(9,12,15,29)(10,18,20,34)(11,23,25,39)(13,28,14,27)(16,30,19,26)(17,31,33,37)(21,35,32,40)(22,36,24,38)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,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, 308273);