Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #61617 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1137 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)(3,7,8,5)(4,6)(9,10)(11,20)(12,22)(13,16)(14,19)(15,18)(17,24)(21,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,18)(14,22)(15,21)(16,24)(17,23)(19,20);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,15)(14,23)(17,21)(18,22)(20,24);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,15)(14,23)(16,19)(17,22)(18,21)(20,24)Polycyclic presentation:
2^9|a^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,e^d=ei,f^d=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180567);