Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(2,6)(5,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^8|a^2=g,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105756, #107160, #107180, … |
| maximal normal subgroup of | #105756, #107160, #107180, … |
| maximal quotient of | #213496, #213497, #213498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14742);