Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| 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 | |
513 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 111 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);(3,7)(5,8)(9,10)(11,16)(12,20)(13,15)(14,17)(18,24)(19,23)(21,22);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,23)(20,24);(11,19)(16,23)(18,21)(22,24)Polycyclic presentation:
2^8|e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152592, #153167, #153169, … |
| maximal normal subgroup of | #152592, #153167, #153169, … |
| maximal quotient of | #167761, #167763, #167797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12038);