Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | Aut(G) = #72338915 (order 4096) · Inn(G) = #2826 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,7,12,8,14,6,13)(3,15,19,33,20,43,18,9)(4,21,25,10,26,36,24,37)(5,11,28,41,29,42,30,40)(16,31,22,34,48,57,47,44)(17,45,50,38,52,59,51,32)(23,39,27,35,53,61,54,62)(46,56,49,60,55,63,64,58);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,43)(4,22,24,16,26,47,25,48)(5,17,28,50,29,52,30,51)(10,34,36,57,37,44,21,31)(11,38,40,45,42,32,41,59)(23,49,27,55,53,64,54,46)(35,58,39,63,62,60,61,56);(1,4,8,26)(2,10,14,37)(3,16,20,48)(5,27,29,54)(6,25,7,24)(9,31,33,57)(11,39,42,61)(12,21,13,36)(15,44,43,34)(17,49,52,64)(18,22,19,47)(23,30,53,28)(32,58,38,60)(35,41,62,40)(45,63,59,56)(46,51,55,50);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,35)(12,41)(13,40)(14,42)(15,45)(16,46)(18,51)(19,50)(20,52)(21,39)(22,49)(24,54)(25,27)(26,53)(31,56)(33,38)(34,60)(36,61)(37,62)(43,59)(44,58)(47,64)(48,55)(57,63)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,d^b=def,f^b=fh,d^c=de,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151237, #151239, #151241, … |
| maximal normal subgroup of | #151237, #151239, #151241, … |
| maximal quotient of | #391210, #391251, #391526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25860);