Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,24)(7,16,19,26,20,30,18,12)(8,13,23,28,21,27,22,34)(17,29,33,35,31,36,32,25);(5,7,9,18,11,20,10,19)(6,12,14,16,15,26,24,30)(8,17,22,32,21,31,23,33)(13,25,28,29,27,35,34,36);(1,2,3,4)(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33);(5,8,10,23,11,21,9,22)(6,13,14,28,15,27,24,34)(7,17,19,33,20,31,18,32)(12,25,16,29,26,35,30,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124457, #124464, #128932, … |
| maximal normal subgroup of | #124457, #124464, #128932, … |
| maximal quotient of | #212927, #212928, #212929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14695);