Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
563 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 163 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 — 4 generators on 24 points:
(2,4)(6,8)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2,3,4)(10,14)(15,23)(16,21)(17,24);(5,6,7,8);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #100504, #100506, #100520, … |
| maximal normal subgroup of | #100504, #100506, #100520, … |
| maximal quotient of | #351655, #351657, #351659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26153);