Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273051 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11,12,10)(8,9)(14,15)(16,18)(19,20);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,14,17,19)(15,16,20,18);(2,4)(5,8,9,6)(7,11,12,10);(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,18,19,16)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=hi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180373);