Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
713 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 713 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(2,5)(6,8)(9,10,12,13)(11,14,16,15)(17,18)(19,20)(21,22)(23,24)(26,29)(30,31);(3,7)(6,8)(10,13)(14,15)(18,22)(19,24)(25,26,28,29)(27,30,32,31);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,29)(30,32);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,13)(14,16)(17,20,21,23)(18,24,22,19)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^8|a^2=eg,b^2=eh,c^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | #144655, C2×#17272, #845126, … |
| maximal normal subgroup of | #144655, C2×#17272, #845126, … |
| maximal quotient of | #156547, #156588, #156589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10260);