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 | #61621 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1120 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)(3,8,5,7)(4,6)(9,10)(11,20)(12,17)(13,22)(14,21)(15,18)(16,19)(23,24);(1,3,6,5)(2,7,4,8)(9,11,16,23)(10,13,19,12)(14,17,15,22)(18,24,21,20);(1,4,6,2)(3,8,5,7)(9,12,16,13)(10,17,19,22)(11,21,23,18)(14,24,15,20);(1,5,6,3)(2,8,4,7)(9,13,16,12)(10,11,19,23)(14,24,15,20)(17,21,22,18)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=h,d^2=h,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, 180620);