Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#459 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,15,17,34)(4,21,24,54)(5,13,7,11)(6,26,29,38)(9,20,32,18)(10,35,37,23)(12,25,40,28)(16,44,45,64)(19,33,48,60)(22,55,30,39)(27,41,57,36)(31,58,59,50)(42,62,61,47)(43,49,56,51)(46,53,52,63);(1,3,5,18,8,17,7,20)(2,9,11,15,14,32,13,34)(4,16,22,46,24,45,30,52)(6,19,25,49,29,48,28,51)(10,31,36,42,37,59,41,61)(12,33,38,43,40,60,26,56)(21,53,39,44,54,63,55,64)(23,47,27,50,35,62,57,58);(1,4)(2,10)(3,16)(5,22)(6,27)(7,30)(8,24)(9,31)(11,36)(12,39)(13,41)(14,37)(15,42)(17,45)(18,46)(19,50)(20,52)(21,26)(23,28)(25,35)(29,57)(32,59)(33,44)(34,61)(38,54)(40,55)(43,63)(47,51)(48,58)(49,62)(53,56)(60,64);(3,17)(9,32)(15,34)(16,45)(18,20)(19,48)(31,59)(33,60)(42,61)(43,56)(44,64)(46,52)(47,62)(49,51)(50,58)(53,63)Polycyclic presentation:
2^8|a^2=h,b^2=e,e^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500647, #500649, #500651, … |
| maximal normal subgroup of | #500647, #500649, #500651, … |
| maximal quotient of | #490435, #490437, #490438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26584);