Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128055 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1234 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)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(1,2)(3,7,8,5)(4,6)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=e,b^2=f,f^2=g,b^a=bd,d^a=dg,f^a=fh,c^b=ch,d^b=dh,e^b=eg,h^b=hi,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231794, #72231819, #72231827, … |
| maximal normal subgroup of | #72231794, #72231819, #72231827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35379);