Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,25,28,26)(24,27);(2,4)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,28)(25,27);(13,16,17,24)(14,19,20,27)(15,22,23,28)(18,25,21,26)Polycyclic presentation:
2^9|a^2=g,c^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^b=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257298, #72257301, #72257318, … |
| maximal normal subgroup of | #72257298, #72257301, #72257318, … |
| maximal quotient of | #72264452 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18754);