Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 437 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 — 3 generators on 64 points:
(1,2,6,13,8,12,7,29)(3,14,17,42,18,41,30,38)(4,19,21,37,22,10,28,36)(5,23,25,39,27,56,26,11)(9,24,33,48,32,57,34,16)(15,43,45,62,47,40,46,61)(20,51,53,50,54,55,58,35)(31,52,59,44,49,63,60,64);(1,3)(2,9)(4,15)(5,24)(6,17)(7,30)(8,18)(10,31)(11,38)(12,32)(13,33)(14,23)(16,26)(19,49)(20,52)(21,45)(22,47)(25,48)(27,57)(28,46)(29,34)(35,61)(36,59)(37,60)(39,42)(40,55)(41,56)(43,51)(44,53)(50,62)(54,63)(58,64);(1,4,8,22)(2,10,12,19)(3,15,18,47)(5,20,27,54)(6,28,7,21)(9,31,32,49)(11,35,39,50)(13,37,29,36)(14,40,41,43)(16,44,48,64)(17,46,30,45)(23,55,56,51)(24,52,57,63)(25,58,26,53)(33,60,34,59)(38,61,42,62)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130777, #130778, #130783, … |
| maximal normal subgroup of | #130777, #130778, #130783, … |
| maximal quotient of | #129517, #129518, #129559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5732);