Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,20,25)(19,28,31,32,30,23,29,24);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,31,21,30,20,29)(18,23,27,24,26,28,25,32);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,19,21,30)(18,24,26,32)(20,31,22,29)(23,25,28,27);(3,7)(6,8)(17,20,21,22)(18,25,26,27)(19,29,30,31)(23,32,28,24)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #170143, #170145, #170175, … |
| maximal normal subgroup of | #170143, #170145, #170175, … |
| maximal quotient of | #380112, #380521, #380965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23178);