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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3844 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)(10,11)(14,15)(17,18)(19,22)(20,23)(21,24);(6,8)(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,23)(22,24);(1,2,4,3)(5,6)(7,8);(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(1,3,4,2)(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,e^2=fg,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 | #72280839, #72281037, #72281096, … |
| maximal normal subgroup of | #72280839, #72281037, #72281096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1210037);