Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,29)(28,32)(30,34)(31,33)(35,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,26)(24,31)(25,30)(27,33)(28,35)(29,34)(32,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(6,12)(10,13)(11,20)(18,19)(23,30)(26,34)(31,36)(33,35)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21843, #6374366, #6374367, … |
| maximal normal subgroup of | C2×#21843, #6374366, #6374367, … |
| maximal quotient of | #215714, #215715, #215716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14831);