Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18,20,21)(19,23,24,22);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,15,14)(17,19)(18,22)(20,24)(21,23);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(18,21)(19,23)(22,24);(3,7)(5,8)(11,16)(13,14)(17,18,20,21)(19,22,24,23)Polycyclic presentation:
2^8|a^2=e,b^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #169382, #169388, #169414, … |
| maximal normal subgroup of | #169382, #169388, #169414, … |
| maximal quotient of | #391236, #391305, #391328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25731);