Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2688 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 97 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)(3,13)(4,18)(5,22)(6,26)(7,12)(8,16)(9,20)(10,24)(11,30)(14,42)(15,46)(17,33)(19,35)(21,40)(23,56)(25,41)(27,39)(28,36)(29,38)(31,48)(32,47)(34,45)(37,49)(43,57)(44,50)(51,53)(52,58)(54,60)(55,59)(61,63)(62,64);(1,3,6,27,7,16,30,17)(2,8,11,39,12,13,26,33)(4,14,20,48,21,44,28,45)(5,23,24,47,25,15,29,49)(9,34,18,50,36,31,40,42)(10,37,22,46,38,32,41,56)(19,52,53,62,54,43,55,63)(35,57,59,64,60,58,51,61);(1,4)(2,9)(3,14,16,44)(5,19)(6,28)(7,21)(8,31,13,34)(10,35)(11,40)(12,36)(15,43,23,52)(17,48,27,45)(18,26)(20,30)(22,51)(24,55)(25,54)(29,53)(32,57,37,58)(33,42,39,50)(38,60)(41,59)(46,64,56,61)(47,63,49,62);(1,5,7,25)(2,10,12,38)(3,15,16,23)(4,19,21,54)(6,29,30,24)(8,32,13,37)(9,35,36,60)(11,41,26,22)(14,43,44,52)(17,49,27,47)(18,51,40,59)(20,55,28,53)(31,57,34,58)(33,56,39,46)(42,61,50,64)(45,63,48,62)Polycyclic presentation:
2^8|b^2=ef,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792406, #6792407, #6792446, … |
| maximal normal subgroup of | #6792406, #6792407, #6792446, … |
| maximal quotient of | #345969, #345970, #345981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24407);