Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,13,18,14,17,26,31)(11,19,22,32,21,15,25,28)(12,16,24,29,23,30,36,40)(20,27,35,38,34,33,37,39);(1,2,4,6)(3,7,8,5)(9,11)(10,15)(12,20)(13,25)(14,21)(16,27)(17,19)(18,32)(22,26)(23,34)(24,37)(28,31)(29,39)(30,33)(35,36)(38,40);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,29)(19,33)(21,34)(22,35)(25,37)(26,36)(28,38)(31,40)(32,39)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #11391⋊C3 |
| maximal subgroup of | #93977, #137311, #137312, … |
| maximal normal subgroup of | #93977, #137311, #137312, … |
| maximal quotient of | #121985, #122004, #122007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4379);