Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(10,13)(11,15)(17,18,21,19)(20,23,24,22);(5,6,8,7)(10,13)(14,16)(18,19)(20,24);(1,3,4,2)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de76 conjugacy classes — view the full character table.
| maximal subgroup of | #139635, C2×#13831, #513764, … |
| maximal normal subgroup of | #139635, C2×#13831, #513764, … |
| maximal quotient of | #153730, #153731, #153969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6819);