Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,14,18,15,19,13,30)(11,20,24,32,22,36,23,16)(12,25,28,34,29,40,27,17)(21,33,38,39,26,31,37,35);(1,2,3,4)(9,11)(10,16,19,32)(12,26)(13,23)(14,24)(15,22)(17,33,34,31)(18,20,30,36)(21,29)(25,39,40,35)(27,38)(28,37);(9,12,13,27,15,29,14,28)(10,17,18,25,19,34,30,40)(11,21,23,37,22,26,24,38)(16,31,20,35,32,33,36,39);(6,8)(11,22)(16,32)(20,36)(21,26)(23,24)(31,33)(35,39)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300205);