Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10549 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #394 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10,14,17)(11,18,20,24)(12,21,16,13)(15,22,23,19);(1,3)(2,4)(5,8)(6,7)(9,11,14,20)(10,15,17,23)(12,18,16,24)(13,22,21,19);(2,7)(3,8)(4,5)(9,12,13,10,14,16,21,17)(11,18,19,23,20,24,22,15)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=d,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,e^c=egh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10554, #240936, #240944, … |
| maximal normal subgroup of | C2×#10554, #240936, #240944, … |
| maximal quotient of | #104887, #104888, #104889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3542);