Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,7,12)(3,8,14,30)(4,17,18,52)(5,21,26,57)(6,27,24,11)(9,33,34,19)(10,36,39,25)(13,41,42,51)(15,46,22,56)(16,49,45,32)(20,38,28,53)(23,35,55,40)(29,54,60,43)(31,59,37,48)(44,62,50,64)(47,61,58,63);(1,3)(2,8)(4,13)(5,22)(6,16)(7,14)(9,29)(10,37)(11,32)(12,30)(15,26)(17,51)(18,42)(19,54)(20,44)(21,56)(23,58)(24,45)(25,59)(27,49)(28,50)(31,39)(33,43)(34,60)(35,61)(36,48)(38,64)(40,63)(41,52)(46,57)(47,55)(53,62);(1,4,6,28,7,18,24,20)(2,9,11,40,12,34,27,35)(3,13,16,50,14,42,45,44)(5,23,25,33,26,55,36,19)(8,29,32,63,30,60,49,61)(10,38,21,52,39,53,57,17)(15,47,48,54,22,58,59,43)(31,62,46,51,37,64,56,41);(3,14)(8,30)(13,42)(15,22)(16,45)(29,60)(31,37)(32,49)(41,51)(43,54)(44,50)(46,56)(47,58)(48,59)(61,63)(62,64)Polycyclic presentation:
2^8|a^2=h,c^2=eg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142698, #142699, #142702, … |
| maximal normal subgroup of | #142698, #142699, #142702, … |
| maximal quotient of | #493921, #494271, #494486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26646);