Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(10,15)(11,19)(12,13)(17,18)(20,21)(22,23);(5,6,7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22);(1,3,4,2)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,e^b=ei,f^c=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261124);