Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 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)(5,6,8,7)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,24)(17,29)(22,31)(23,33)(25,32)(27,34)(28,36)(30,35)(37,39)(38,40);(9,11)(10,16,20,32)(12,27)(13,24)(14,25)(15,23)(17,34,36,31)(18,21,19,33)(22,30)(26,37,35,40)(28,39)(29,38);(2,4)(9,12,13,28,15,30,14,29)(10,17,18,35,20,36,19,26)(11,22,24,38,23,27,25,39)(16,31,21,37,32,34,33,40);(5,7,8,6)(11,23)(16,32)(21,33)(22,27)(24,25)(31,34)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261560);