Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,15,12,16,10,17)(7,8,18,13,9,20,19,14);(1,2)(3,4)(5,7)(6,13)(8,17)(9,12)(10,19)(11,18)(14,16)(15,20);(2,4)(5,8,11,13,12,20,10,14)(6,9,15,19,16,7,17,18);(5,9)(6,14)(7,12)(8,15)(10,18)(11,19)(13,16)(17,20)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322946 |
| maximal normal subgroup of | #72322946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280878);