Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6,8,7)(9,10)(11,19)(12,15)(13,22)(14,16)(17,20)(18,23)(21,24);(1,2)(3,4)(5,7,8,6)(9,11,13,12,14,20,23,21)(10,15,18,19,16,24,22,17);(2,4)(6,7)(10,16)(11,20)(12,21)(18,22);(9,12,13,20,14,21,23,11)(10,17,18,15,16,19,22,24)Polycyclic presentation:
2^8|a^2=fh,b^2=efh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #502208, #504479, #504843, … |
| maximal normal subgroup of | #502208, #504479, #504843, … |
| maximal quotient of | #288366, #288372, #288378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18415);