Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,21,23,20);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,23)(18,20,24,21);(2,5)(6,7)(10,13)(14,15)(17,18)(19,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,19,24,23);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #133470, #254474, #254476, … |
| maximal normal subgroup of | #133470, #254474, #254476, … |
| maximal quotient of | #515501, #515505, #515655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28427);