Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,17,14,18,12,16)(11,15,21,24,19,22,20,23);(2,5)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22);(11,19)(15,22)(20,21)(23,24);(9,12,14,13)(10,16,18,17)(11,20,19,21)(15,23,22,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=chi,f^b=fg,f^e=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387781);