Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,5,6,7)(17,19)(18,21)(20,23)(22,24);(10,14)(11,12)(15,16)(21,24);(9,11,13,12)(10,14,16,15)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282161, #72282194, #72282276, … |
| maximal normal subgroup of | #72282161, #72282194, #72282276, … |
| maximal quotient of | C2×#237802 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144998);