Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6,8,7)(10,11)(14,15)(17,18,20,21)(19,22,23,24);(5,7,8,6)(9,10,13,11)(12,14,16,15)(18,21)(19,22)(23,24);(6,7)(9,11,13,10)(12,14,16,15)(17,19,20,23)(18,22,21,24);(9,12,13,16)(10,14,11,15)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287396);