Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1061 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)(11,16)(12,14)(13,15)(17,18,23,24)(19,20,22,21);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,22)(18,21,24,20);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,20,23,21)(18,19,24,22);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,23,20)(18,19,24,22);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,22,23,19)(18,21,24,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132287, #253826, #253828, … |
| maximal normal subgroup of | #132287, #253826, #253828, … |
| maximal quotient of | #517989, #517991, #518004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28920);