Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,20,25,17,29,19,12)(8,13,22,27,23,28,21,34)(18,30,33,35,31,36,32,26);(5,7,9,19,11,17,10,20)(6,12,14,16,15,25,24,29)(8,18,21,32,23,31,22,33)(13,26,27,30,28,35,34,36);(2,4)(7,17)(12,25)(16,29)(18,31)(19,20)(26,35)(30,36)(32,33);(5,8,10,22,11,23,9,21)(6,13,14,27,15,28,24,34)(7,18,20,33,17,31,19,32)(12,26,16,30,25,35,29,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #124166, #128773, #134422, … |
| maximal normal subgroup of | #124166, #128773, #134422, … |
| maximal quotient of | C8×#468, C8×#469, C8×#470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14536);