Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,12,17)(11,20,21,15)(13,16,24,32)(14,27,25,19)(18,34,33,26)(22,28,37,31)(23,30,38,36)(29,40,35,39);(1,3,5,8)(2,6,4,7)(9,11,12,21)(10,15,17,20)(13,22,24,37)(14,23,25,38)(16,31,32,28)(18,29,33,35)(19,30,27,36)(26,40,34,39);(10,16)(11,21)(13,26)(14,25)(15,28)(17,32)(18,19)(20,31)(22,39)(24,34)(27,33)(29,36)(30,35)(37,40);(1,4,5,2)(3,6,8,7)(11,21)(15,20)(22,37)(23,38)(28,31)(29,35)(30,36)(39,40)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33955, #7623924, #7623926, … |
| maximal normal subgroup of | C2×#33955, #7623924, #7623926, … |
| maximal quotient of | #497616, #497617, #497873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26943);