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 | #61620 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1096 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)(3,8,5,7)(4,6)(9,10,15,19)(11,20,17,24)(12,16,22,21)(13,23,14,18);(1,3,6,5)(2,7,4,8)(9,11,15,17)(10,12,19,22)(13,16,14,21)(18,24,23,20);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,18)(13,20)(14,24)(15,22)(17,23)(19,21);(1,5,6,3)(2,8,4,7)(9,12)(10,17)(11,19)(13,24)(14,20)(15,22)(16,23)(18,21)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=hi,d^2=hi,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, 180566);