Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(2,4)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,2,3,4)(5,6)(7,8);(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97833, #97835, #97841, … |
| maximal normal subgroup of | #97833, #97835, #97841, … |
| maximal quotient of | #115700, #115726, #115728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4033);