Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,12,15)(11,14,16,13)(17,18,20,22)(19,23,24,21);(3,8)(4,6)(9,11)(10,13)(12,16)(14,15);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,14,15,13)(17,19,20,24)(18,21,22,23);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=gh,c^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133626, #500983, #503375, … |
| maximal normal subgroup of | #133626, #500983, #503375, … |
| maximal quotient of | #560566, #561674, #563278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41827);