Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,17)(9,11)(10,15)(12,14)(16,19)(18,20)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(22,26)(27,34)(28,33)(29,35);(6,11)(10,18)(12,17)(13,19)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35)Polycyclic presentation:
2^8|a^2=f,b^2=dgh,c^2=eg,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #8817⋊C3 |
| maximal subgroup of | #93084, #135095, #135096, … |
| maximal normal subgroup of | #93084, #135095, #135096, … |
| maximal quotient of | #99299, #99305, #109003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1805);