Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,13,10,14,20,26)(7,11,17,23,16,22,30,34)(8,12,19,25,18,24,31,35)(15,27,29,36,28,21,32,33);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,23)(14,22)(18,28)(19,29)(20,30)(24,27)(25,33)(26,34)(31,32)(35,36);(5,8,9,19,10,18,20,31)(6,12,13,25,14,24,26,35)(7,15,17,29,16,28,30,32)(11,21,23,33,22,27,34,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,f^2=h,b^a=bd,c^a=ce,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10920, #353985, #354316, … |
| maximal normal subgroup of | C2×#10920, #353985, #354316, … |
| maximal quotient of | #109582, #109583, #109584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3908);