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 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,13)(11,12,16,15)(19,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,13)(14,16)(17,18,21,20)(19,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,12,16,15)(17,20,21,18)(19,24,23,22);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,13)(14,15)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129964, #253249, #253251, … |
| maximal normal subgroup of | #129964, #253249, #253251, … |
| maximal quotient of | #517645, #517649, #517672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28827);