Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,23)(22,24);(1,2,3,4)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(18,21)(19,24)(22,23);(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239676, #72239688, #72239783, … |
| maximal normal subgroup of | #72239676, #72239688, #72239783, … |
| maximal quotient of | #72250406 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308763);