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)(11,19)(12,13)(14,17)(15,18)(16,21)(20,22)(23,24);(5,7,8,6)(9,11,14,22,15,23,13,21)(10,16,12,24,18,20,17,19);(6,7)(9,12,13,10,15,17,14,18)(11,20,21,24,23,16,22,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,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 | #136199, #136200, #136201, … |
| maximal normal subgroup of | #136199, #136200, #136201, … |
| maximal quotient of | #120605, #120607, #120610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4258);