Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10,13,12,14,16,21,20)(11,15,19,17,18,22,24,23);(2,4)(5,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24);(9,12,13,16,14,20,21,10)(11,17,19,22,18,23,24,15);(11,18)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,f^2=h,b^a=be,c^a=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #108487, #108633, #109703, … |
| maximal normal subgroup of | #108487, #108633, #109703, … |
| maximal quotient of | #183284, #183290, #183297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13426);