Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20)(21,22,27,23)(24,26,34,30)(25,35,36,31)(28,33,29,32);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19)(21,23,26,35,28,33,25,34,27,22,30,31,29,32,36,24);(6,11)(10,18)(12,16)(13,19)(21,24,25,33,29,31,26,23,27,34,36,32,28,35,30,22)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=eg,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42968);