Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2644 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,28)(3,14,29,9,18,32,17,33)(4,10,21,36,22,37,23,41)(5,11,25,39,26,40,27,52)(15,34,48,56,44,55,19,30)(16,42,53,31,47,58,46,59)(20,35,24,38,49,61,50,62)(43,60,63,64,51,57,45,54);(1,3,7,18)(2,9)(4,19,22,48)(5,16,26,47)(6,17,8,29)(10,34)(11,31)(12,14)(13,33)(15,21,44,23)(20,45,49,63)(24,51,50,43)(25,46,27,53)(28,32)(30,36)(35,60)(37,55)(38,54)(39,42)(40,59)(41,56)(52,58)(57,61)(62,64);(1,4,7,22)(2,10,13,37)(3,15,18,44)(5,24,26,50)(6,23,8,21)(9,30,33,56)(11,38,40,62)(12,41,28,36)(14,34,32,55)(16,45,47,63)(17,48,29,19)(20,27,49,25)(31,57,59,60)(35,52,61,39)(42,54,58,64)(43,53,51,46);(1,5,8,27,7,26,6,25)(2,11,28,52,13,40,12,39)(3,16,29,53,18,47,17,46)(4,20,23,50,22,49,21,24)(9,31,32,58,33,59,14,42)(10,35,41,62,37,61,36,38)(15,43,48,63,44,51,19,45)(30,54,34,60,56,64,55,57)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=de,e^c=eh,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #30238⋊C3 |
| maximal subgroup of | #507895, #507896, #508806, … |
| maximal normal subgroup of | #507895, #507896, #508806, … |
| maximal quotient of | #380191, #380395, #382344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23226);