Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,3,4)(5,6,8,7)(9,10,14,17)(11,18,23,24)(12,19,16,13)(15,22,20,21);(5,7,8,6)(9,11,19,22,14,23,13,21)(10,15,12,24,17,20,16,18);(6,7)(9,12,13,10,14,16,19,17)(11,20,21,24,23,15,22,18)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93948, #136222, C2×#11271, … |
| maximal normal subgroup of | #93948, #136222, C2×#11271, … |
| maximal quotient of | #120606, #120608, #120609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4259);