Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106570 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,15)(12,13)(16,23)(17,18)(19,21)(22,24);(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,12,15)(11,16,19,23,20,24,21,22);(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214287, #72214750, #72214788, … |
| maximal normal subgroup of | #72214287, #72214750, #72214788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13768);