Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21)(11,16,25,38)(12,17,29,41)(13,30,32,59)(14,33,31,20)(18,42,44,26)(19,45,46,27)(22,34,51,62)(23,52,54,72)(24,55,53,37)(28,47,56,40)(35,63,65,48)(36,66,67,49)(39,58,43,57)(50,68,69,61)(60,71,64,70);(2,5)(6,8)(9,11,15,25)(10,16,21,38)(12,22,29,51)(13,23,32,54)(14,24,31,53)(17,34,41,62)(18,35,44,65)(19,36,46,67)(20,37,33,55)(26,48,42,63)(27,49,45,66)(28,50,56,69)(30,52,59,72)(39,60,43,64)(40,61,47,68)(57,70,58,71);(1,3,4,7)(2,6,5,8)(10,17)(12,26)(13,27)(14,31)(16,34)(18,19)(20,47)(21,41)(22,48)(23,49)(24,53)(28,58)(29,42)(30,39)(32,45)(33,40)(35,36)(37,68)(38,62)(43,59)(44,46)(50,71)(51,63)(52,60)(54,66)(55,61)(56,57)(64,72)(65,67)(69,70)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13716, #512979, #512981, … |
| maximal normal subgroup of | C2×#13716, #512979, #512981, … |
| maximal quotient of | #152366, #152419, #152503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6704);