Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | Aut(G) = C2×#72240903 (order 12288) · Inn(G) = #2846 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | 2−1+6 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1756 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,36)(4,20,23,38)(5,11,25,35)(6,13,7,12)(9,26,31,18)(10,28,33,22)(16,29,44,55)(17,45,48,40)(21,50,53,57)(24,37,27,34)(30,47,56,46)(32,51,58,52)(39,42,49,43)(41,60,62,54)(59,61,64,63);(1,3)(2,9)(4,16)(5,17)(6,26)(7,18)(8,19)(10,29)(11,30)(12,36)(13,15)(14,31)(20,49)(21,41)(22,43)(23,44)(24,47)(25,48)(27,46)(28,42)(32,54)(33,55)(34,40)(35,56)(37,45)(38,39)(50,64)(51,63)(52,61)(53,62)(57,59)(58,60);(1,4,8,23)(2,10,14,33)(3,16,19,44)(5,21,25,53)(6,22,7,28)(9,29,31,55)(11,32,35,58)(12,20,13,38)(15,39,36,49)(17,41,48,62)(18,42,26,43)(24,51,27,52)(30,54,56,60)(34,57,37,50)(40,59,45,64)(46,61,47,63);(1,5)(2,11)(3,17)(4,21)(6,27)(7,24)(8,25)(9,30)(10,32)(12,37)(13,34)(14,35)(15,40)(16,41)(18,47)(19,48)(20,50)(22,52)(23,53)(26,46)(28,51)(29,54)(31,56)(33,58)(36,45)(38,57)(39,59)(42,63)(43,61)(44,62)(49,64)(55,60)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,g^b=gh,f^c=fh,e^d=eh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #32897⋊C3, #72341975, #72342010 |
| maximal subgroup of | #499822, #499823, #499826, … |
| maximal normal subgroup of | #499822, #499823, #499826, … |
| maximal quotient of | #346493, #346494, #346495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25885);