Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1196 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)(19,24)(20,22)(21,23);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,21,20)(18,22,23,24);(2,5)(6,7)(10,13)(14,15)(19,20)(22,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,22,23,24);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^8|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.
| Fitting subgroup of | #35438⋊C3, #72341122 |
| maximal subgroup of | #133478, #254467, #254469, … |
| maximal normal subgroup of | #133478, #254467, #254469, … |
| maximal quotient of | #515497, #515658, #516171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28426);