Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22,24,27,25,28,32,35)(23,29,31,36,30,26,33,34);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23)(22,26,28,29)(24,31)(25,30)(27,34,35,36)(32,33);(1,2,3,4)(7,14)(10,18)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12618);