Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| 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 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ 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:
(1,2)(3,4)(5,6,9,13,10,14,21,27)(7,15,18,29,17,11,22,24)(8,12,19,25,20,26,32,35)(16,28,31,36,30,23,33,34);(5,7)(6,11,14,15)(8,16)(9,18)(10,17)(12,23,26,28)(13,24,27,29)(19,31)(20,30)(21,22)(25,34,35,36)(32,33);(2,4)(5,8,9,19,10,20,21,32)(6,12,13,25,14,26,27,35)(7,16,18,31,17,30,22,33)(11,23,24,34,15,28,29,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,f^2=h,b^a=bd,c^a=ce,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10914, #353972, #354182, … |
| maximal normal subgroup of | C2×#10914, #353972, #354182, … |
| maximal quotient of | #109566, #109567, #109568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3902);