Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(14,19)(16,17)(20,27)(21,22)(23,25)(26,28);(5,7,8,6,9,10,12,11)(13,14,17,22,18,21,16,19)(15,20,23,27,24,28,25,26);(1,3,4,2)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16891);