Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(5,6)(7,8)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(1,2,3,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265095, #72265123, #72265157, … |
| maximal normal subgroup of | #72265095, #72265123, #72265157, … |
| maximal quotient of | #72279608, #72304146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31059);