Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,34,40,35,46,33,25)(28,41,44,47,42,48,43,37);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,31,23,29,21,30)(18,24,26,39,27,38,36,45)(20,28,34,44,35,42,33,43)(25,37,32,41,40,47,46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,33,23,35,22,34)(18,25,26,32,27,40,36,46)(19,28,30,43,29,42,31,44)(24,37,39,41,38,47,45,48);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139130, #139131, #139136, … |
| maximal normal subgroup of | #139130, #139131, #139136, … |
| maximal quotient of | #219264, #219265, #219266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15246);