Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) = #72316158 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 157 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:
(1,2,6,13,8,11,7,28)(3,14,17,41,18,40,29,38)(4,10,21,35,19,36,22,51)(5,23,25,37,27,53,26,12)(9,24,32,45,31,54,33,16)(15,39,44,60,42,61,52,58)(20,47,49,59,46,63,50,34)(30,48,55,62,57,64,56,43);(1,3)(2,9)(4,15)(5,24)(6,17)(7,29)(8,18)(10,30)(11,31)(12,38)(13,32)(14,23)(16,26)(19,42)(20,48)(21,44)(22,52)(25,45)(27,54)(28,33)(34,58)(35,55)(36,57)(37,41)(39,47)(40,53)(43,50)(46,64)(49,62)(51,56)(59,60)(61,63);(1,4,6,21,8,19,7,22)(2,10,13,35,11,36,28,51)(3,15,17,44,18,42,29,52)(5,20,25,49,27,46,26,50)(9,30,32,55,31,57,33,56)(12,34,23,47,37,59,53,63)(14,39,41,60,40,61,38,58)(16,43,24,48,45,62,54,64);(2,11)(4,19)(9,31)(12,37)(13,28)(14,40)(15,42)(20,46)(21,22)(23,53)(32,33)(38,41)(43,62)(44,52)(48,64)(49,50)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=gh,f^2=h,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #152287, #152288, #152440, … |
| maximal normal subgroup of | #152287, #152288, #152440, … |
| maximal quotient of | #483504, #483505, #483506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26453);