Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 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)(9,10,15,21,16,22,14,20)(11,23,26,17,27,35,25,36)(12,18,30,38,24,39,29,37)(13,31,32,34,33,40,28,19);(9,11,16,27)(10,17,22,36)(12,28,24,32)(13,30,33,29)(14,26,15,25)(18,31,39,40)(19,37,34,38)(20,35,21,23);(2,4)(9,12,16,24)(10,18,22,39)(11,13,27,33)(14,29,15,30)(17,34,36,19)(20,37,21,38)(23,31,35,40)(25,28,26,32);(2,4)(9,13)(10,19)(11,24)(12,27)(14,32)(15,28)(16,33)(17,18)(20,31)(21,40)(22,34)(23,38)(25,29)(26,30)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290252);