Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
755 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 755 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,14,18)(11,19,21,16)(12,15,22,24)(13,23,20,17);(5,6)(7,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(1,2,3,4)(6,8)(11,21)(12,22)(15,24)(16,19);(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142641, #142643, #142645, … |
| maximal normal subgroup of | #142641, #142643, #142645, … |
| maximal quotient of | #298662, #298669, #298672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18774);