Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2156 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)(5,6)(7,8)(9,10,16,22)(11,23,12,27)(13,29,33,37)(14,21,15,20)(17,26,18,25)(19,31,38,32)(24,35,28,34)(30,36,39,40);(9,11,16,12)(10,17,22,18)(13,24,33,28)(14,26,15,25)(19,34,38,35)(20,27,21,23)(29,40,37,36)(30,32,39,31);(2,4)(9,12,16,11)(10,18,22,17)(13,30,33,39)(14,25,15,26)(19,36,38,40)(20,23,21,27)(24,31,28,32)(29,34,37,35);(6,8)(9,13)(10,19)(11,24)(12,28)(14,32)(15,31)(16,33)(17,34)(18,35)(20,29)(21,37)(22,38)(23,36)(25,30)(26,39)(27,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,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,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 361304);