Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
517 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24)(26,27)(30,31);(10,12)(14,15)(19,23)(21,24)(25,26,29,27)(28,31,32,30);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,21)(20,23)(22,24)(25,27,29,26)(28,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #144608, C2×#16171, #610300, … |
| maximal normal subgroup of | #144608, C2×#16171, #610300, … |
| maximal quotient of | #154860, #154975, #155035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9159);