Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19304 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20);(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17);(2,5)(3,8)(6,7);(2,6)(5,7)(11,19)(14,23)(17,24)(20,22)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19307, #6353963, #6353992, … |
| maximal normal subgroup of | C2×#19307, #6353963, #6353992, … |
| maximal quotient of | #189122, #197805, #197806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12295);