Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14)(15,17)(16,22)(18,23)(19,21)(20,25)(24,27)(26,28);(1,2,3,4)(5,7,8,12,9,6,10,11)(13,15,17,14,19,23,18,21)(16,22,24,20,26,28,25,27);(13,16,17,24,19,26,18,25)(14,20,23,28,21,27,15,22);(13,17,19,18)(14,15,21,23)(16,24,26,25)(20,22,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72321367, #72321373, #72321382, … |
| maximal normal subgroup of | #72321367, #72321373, #72321382, … |
| maximal quotient of | #72322546 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111462);