Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2622 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 101 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)(3,14)(4,20)(5,11)(6,28)(7,12)(8,13)(9,17)(10,23)(15,43)(16,41)(18,32)(19,42)(21,30)(22,51)(24,35)(25,39)(26,37)(27,38)(29,36)(31,48)(33,45)(34,53)(40,46)(44,63)(47,60)(49,58)(50,59)(52,56)(54,61)(55,62)(57,64);(1,3,6,18,8,17,7,19)(2,9,12,32,13,14,28,42)(4,21,23,15,25,46,24,45)(5,16,26,49,27,48,29,50)(10,33,20,40,35,43,39,30)(11,31,36,58,38,41,37,59)(22,52,53,44,54,64,55,47)(34,60,51,57,62,63,61,56);(1,4,8,25)(2,10,13,35)(3,15,17,45)(5,22,27,54)(6,24,7,23)(9,30,14,40)(11,34,38,62)(12,39,28,20)(16,47,48,44)(18,21,19,46)(26,55,29,53)(31,57,41,56)(32,43,42,33)(36,61,37,51)(49,64,50,52)(58,60,59,63);(1,5,8,27)(2,11,13,38)(3,16,17,48)(4,22,25,54)(6,26,7,29)(9,31,14,41)(10,34,35,62)(12,36,28,37)(15,44,45,47)(18,49,19,50)(20,51,39,61)(21,52,46,64)(23,53,24,55)(30,56,40,57)(32,58,42,59)(33,60,43,63)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,e^b=eh,d^c=de,f^c=fh,g^c=gh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #500253, #500255, #500260, … |
| maximal normal subgroup of | #500253, #500255, #500260, … |
| maximal quotient of | #330926, #330928, #330931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22893);