Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 413 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,19)(13,16)(15,18)(17,20)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(10,16,18,11)(17,23,24,19);(1,4,5,3)(2,6,7,8)(9,11,13,18,15,16,14,10)(12,19,20,24,22,23,21,17);(2,7)(6,8)(9,12,14,21,15,22,13,20)(10,17,16,23,18,24,11,19)Polycyclic presentation:
2^8|b^2=egh,c^2=e,d^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151354, #151356, #151472, … |
| maximal normal subgroup of | #151354, #151356, #151472, … |
| maximal quotient of | #325967, #325969, #325971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21926);