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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
506 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 506 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)(10,15)(11,18)(12,13)(17,20)(19,21)(22,24);(1,2,4,6)(3,8,7,5)(9,10)(11,16)(12,20)(13,15)(14,17)(18,24)(19,23)(21,22);(3,7)(5,8)(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,23)(20,24);(2,6)(5,8)(10,17)(15,20)(16,23)(22,24)Polycyclic presentation:
2^8|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 | #250780, #250782, #250783, … |
| maximal normal subgroup of | #250780, #250782, #250783, … |
| maximal quotient of | #166305, #166370, #166401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12014);