Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,19,21,24,29,26)(20,23,28,25,27,30,32,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,24,21,26,29,18)(20,25,28,30,27,31,32,23);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(2,6)(5,7)(18,24)(19,26)(20,27)(28,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22294, #6376634, #6376636, … |
| maximal normal subgroup of | C2×#22294, #6376634, #6376636, … |
| maximal quotient of | #355227, #355366, #355368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15282);