Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
376 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)(4,7)(6,8)(9,10,13,15)(11,17,20,24)(12,21,22,16)(14,18,23,19);(9,11,13,20)(10,14,15,23)(12,18,22,19)(16,24,21,17);(2,5)(3,8)(6,7);(10,15)(11,18)(12,22)(14,17)(19,20)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=de,e^b=eg,f^c=fh,e^d=eg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19970, D8×#135, #6358398, … |
| maximal normal subgroup of | C2×#19970, D8×#135, #6358398, … |
| maximal quotient of | SD16×#102, #220803, #220806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12958);