Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 131072 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1099 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)(11,12)(13,14)(15,16)(17,18,20,22)(19,23,24,21);(1,2)(3,7)(4,5)(6,8)(9,11,13,15)(10,12,14,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,7)(10,14)(11,15)(17,19,20,24)(18,21,22,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,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 | #98779, #129486, #134004, … |
| maximal normal subgroup of | #98779, #129486, #134004, … |
| maximal quotient of | #515301, #515336, #515368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28358);