Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 151 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)(3,4)(5,6,8,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(2,4)(5,7,8,6)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #97170, #97171, #97172, … |
| maximal normal subgroup of | #97170, #97171, #97172, … |
| maximal quotient of | #109464, #109466, #109471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3867);