Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(10,12)(13,14,16,21,17,19,25,18)(15,22,23,26,24,20,27,28);(1,3,4,2)(14,18)(15,23)(16,25)(19,21)(24,27)(26,28);(5,6,7,9,8,10,11,12)(14,19)(18,21)(20,22)(26,28);(13,15,16,23,17,24,25,27)(14,20,21,28,19,22,18,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=de,e^b=eh,g^b=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72309439, #72309444, #72309453, … |
| maximal normal subgroup of | #72309439, #72309444, #72309453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138769);