Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
798 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18)(19,22)(20,24)(21,23);(2,3)(6,7)(10,13)(14,16)(17,19,21,20)(18,22,23,24);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,22,23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^8|b^2=fg,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98113, #179362, #179367, … |
| maximal normal subgroup of | #98113, #179362, #179367, … |
| maximal quotient of | #606744, #606792, #606797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46197);