Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246079 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,15,16,13)(18,22)(19,20)(23,24);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(18,23)(19,20)(22,24);(13,15)(17,18)(19,23)(20,22)(21,24);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ehi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72307994, #72308044, #72308058, … |
| maximal normal subgroup of | #72307994, #72308044, #72308058, … |
| maximal quotient of | C2×#265779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172975);