Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #482 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2073 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 47 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)(3,15,19,9)(4,20,24,44)(5,12)(6,13)(7,14)(8,11)(10,22,36,31)(16,47)(17,43,30,33)(18,46,52,34)(21,54)(23,41,57,64)(25,39)(26,58,59,40)(27,60,61,37)(28,42)(29,38)(32,48)(35,55)(45,49)(50,62)(51,63)(53,56);(1,3)(2,9)(4,21)(5,17)(6,26)(7,30)(8,19)(10,35)(11,15)(12,33)(13,40)(14,43)(16,23)(18,28)(20,53)(22,55)(24,56)(25,52)(27,48)(29,59)(31,51)(32,37)(34,42)(36,63)(38,58)(39,46)(41,62)(44,54)(45,60)(47,64)(49,61)(50,57);(1,4)(2,10,11,36)(3,16)(5,22)(6,27)(7,31)(8,24)(9,32,15,45)(12,20,14,44)(13,41,38,64)(17,48)(18,51)(19,50)(21,26)(23,28)(25,57)(29,61)(30,49)(33,62,43,47)(34,54,46,53)(35,58,63,40)(37,39,60,42)(52,55)(56,59)Polycyclic presentation:
2^8|a^2=eg,c^2=d,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13489, #499366, #499595, … |
| maximal normal subgroup of | C2×#13489, #499366, #499595, … |
| maximal quotient of | #146048, #146049, #146055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6477);