Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#3186 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(14,18)(15,16)(19,20);(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20);(5,7)(6,9)(8,12)(10,11)(14,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313309, #72313311, #72313421, … |
| maximal normal subgroup of | #72313309, #72313311, #72313421, … |
| maximal quotient of | #72314207, #72314219 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18683);