Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18,20,22)(19,21,23,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(19,23)(21,24);(9,11,13,12)(10,15,16,14)(17,19,20,23)(18,21,22,24);(2,7)(3,8)(9,12,13,11)(10,14,16,15);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #98766, #165804, #165805, … |
| maximal normal subgroup of | #98766, #165804, #165805, … |
| maximal quotient of | #520775, #520779, #520817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45260);