Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| 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 | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,25)(10,14)(11,15)(12,18)(13,23)(17,32)(19,28)(20,27)(22,30)(24,29)(26,34)(31,33)(35,36);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,23,33,24,34,22,35)(13,26,29,36,30,32,21,31);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,34)(21,25)(27,36)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10900, #353930, C4×#1881, … |
| maximal normal subgroup of | C2×#10900, #353930, C4×#1881, … |
| maximal quotient of | #109522, #109523, #109528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3888);