Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,11)(14,15)(17,18,20,22)(19,23,24,21);(2,3)(6,7)(10,11)(12,16);(9,10,13,11)(12,14,16,15)(17,19,20,24)(18,21,22,23);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^8|a^2=fgh,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97189, #97329, #130298, … |
| maximal normal subgroup of | #97189, #97329, #130298, … |
| maximal quotient of | #557112, #557407, #557791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39140);