Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(5,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24);(10,16)(14,23)(15,19)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232805, #72233110, #72233156, … |
| maximal normal subgroup of | #72232805, #72233110, #72233156, … |
| maximal quotient of | #72232798, #72233000, #72236961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145278);