Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,21)(22,24);(6,8)(10,14)(12,16)(18,21)(22,23);(5,6)(7,8)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(9,11,13,15)(10,12,14,16)(17,19)(18,22)(20,24)(21,23);(9,12)(10,15)(11,14)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=fg,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72323305, #72323331, #72323363, … |
| maximal normal subgroup of | #72323305, #72323331, #72323363, … |
| maximal quotient of | #72323554, #72329054, C2×#647799 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 554995);