Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(18,22)(19,20)(23,24);(10,14)(13,15)(17,18)(19,22)(20,24)(21,23);(1,4,5,3)(2,6,7,8);(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257862, #72257864, #72257879, … |
| maximal normal subgroup of | #72257862, #72257864, #72257879, … |
| maximal quotient of | #72307217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339854);