Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator | C22 | C43 |
| socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,12,9,11,15,18)(7,10,14,16,13,17,19,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(1,2,3,4)(6,11)(7,13)(12,18)(14,19)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,c^a=cgh,c^b=cg112 conjugacy classes — view the full character table.
| Fitting subgroup of | #7845⋊C3 |
| maximal subgroup of | #128230, #128232, #135637, … |
| maximal normal subgroup of | #128230, #128232, #135637, … |
| maximal quotient of | #94854, #94856, #94859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 833);