Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
474 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,11)(12,19,20,24)(13,18,23,16)(15,22,17,21);(1,2,4,6)(3,8,7,5)(9,11,14,10)(12,17,20,15)(13,16,23,18)(19,22,24,21);(3,7)(5,8)(9,12)(10,15)(11,17)(13,22)(14,20)(16,24)(18,19)(21,23);(2,6)(5,8)Polycyclic presentation:
2^8|a^2=gh,b^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250844, #250846, #250847, … |
| maximal normal subgroup of | #250844, #250846, #250847, … |
| maximal quotient of | #166314, #166361, #166410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12021);