Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = C4×#511819 (order 2048) · Inn(G) = C22×D4 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×C8×D4 |
| prime radical | #575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 197 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:
(1,2,6,12,7,11,26,36,8,13,27,37,5,23,25,49)(3,14,17,39,16,9,28,30,18,31,43,55,24,32,50,56)(4,10,21,35,22,34,48,58,19,33,44,57,20,45,47,61)(15,38,42,59,41,29,51,53,40,54,60,64,46,52,62,63);(1,3)(2,9)(4,15)(5,24)(6,17)(7,16)(8,18)(10,29)(11,31)(12,30)(13,32)(14,23)(19,40)(20,46)(21,42)(22,41)(25,50)(26,28)(27,43)(33,52)(34,54)(35,53)(36,55)(37,56)(38,45)(39,49)(44,60)(47,62)(48,51)(57,63)(58,64)(59,61);(1,4,6,21,7,22,26,48,8,19,27,44,5,20,25,47)(2,10,12,35,11,34,36,58,13,33,37,57,23,45,49,61)(3,15,17,42,16,41,28,51,18,40,43,60,24,46,50,62)(9,29,30,53,31,54,55,64,32,52,56,63,14,38,39,59);(4,19)(10,33)(15,40)(20,22)(21,44)(29,52)(34,45)(35,57)(38,54)(41,46)(42,60)(47,48)(51,62)(53,63)(58,61)(59,64)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=h,f^2=g,g^2=h,b^a=be,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #150107, #150108, #491120, … |
| maximal normal subgroup of | #150107, #150108, #491120, … |
| maximal quotient of | #483514, #483527, #483528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26430);