Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
363 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 61 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 — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,12,16)(11,19,20,23,22,24,21,15);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,18,23)(12,21)(13,20)(14,22)(16,24,17,19);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132901, #132902, #132905, … |
| maximal normal subgroup of | #132901, #132902, #132905, … |
| maximal quotient of | #100164, #100166, #100172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2468);