Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,9,11,8,7,10,12)(13,14,16,23,19,22,27,28,20,26,17,24,18,25,15,21);(1,2,3,4)(5,7,9,12,8,6,10,11)(14,21)(15,16)(17,27)(18,19)(22,24)(23,25)(26,28);(6,7)(11,12)(13,15)(14,22)(16,18)(17,19)(20,27)(21,28)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=eg,f^c=fg,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72321045, #72323048, #72323077, … |
| maximal normal subgroup of | #72321045, #72323048, #72323077, … |
| maximal quotient of | #72321050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43915);