Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #392 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,27)(19,29)(20,28)(22,30)(23,32)(25,31)(33,35)(34,36);(5,7,9,18,11,20,10,19)(6,12,14,16,15,29,26,28)(8,22,23,33,25,34,24,17)(13,30,31,27,32,36,21,35);(5,8,9,24,11,25,10,23)(6,13,14,21,15,32,26,31)(7,17,19,22,20,33,18,34)(12,27,28,36,29,35,16,30)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #11337⋊C3 |
| maximal subgroup of | #93954, #136136, #136137, … |
| maximal normal subgroup of | #93954, #136136, #136137, … |
| maximal quotient of | #121328, #121341, #121342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4325);