Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C2×#2540 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24);(6,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,23,20,24);(1,2,4,3)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,23,21,24);(9,11)(10,15)(12,14)(13,16)(17,19,22,21)(18,24,20,23);(1,3,4,2)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^9|b^2=fi,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280828, #72281029, #72281096, … |
| maximal normal subgroup of | #72280828, #72281029, #72281096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1210035);