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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2034 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 53 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)(3,13)(4,19)(5,24)(6,28)(7,12)(8,15)(9,22)(10,27)(11,26)(14,42)(16,32)(17,33)(18,34)(20,53)(21,55)(23,52)(25,38)(29,37)(30,51)(31,47)(35,58)(36,57)(39,56)(40,44)(41,48)(43,64)(45,50)(46,59)(49,63)(54,61)(60,62);(1,3)(2,8)(4,20)(5,17)(6,16)(7,18)(9,31)(10,34)(11,32)(12,33)(13,24)(14,23)(15,27)(19,50)(21,43)(22,54)(25,45)(26,48)(28,59)(29,60)(30,49)(35,40)(36,62)(37,61)(38,42)(39,63)(41,51)(44,57)(46,56)(47,58)(52,64)(53,55);(1,4,7,23)(2,9,12,36)(3,14)(5,25,27,21)(6,29,30,58)(8,31)(10,37,24,35)(11,38,39,55)(13,40)(15,45)(16,47)(17,43)(18,20)(19,51,52,28)(22,56,57,26)(32,42)(33,62)(34,61)(41,64)(44,48)(46,54)(49,60)(50,59)(53,63)Polycyclic presentation:
2^8|c^2=dg,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dh,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13401, #499037, #499595, … |
| maximal normal subgroup of | C2×#13401, #499037, #499595, … |
| maximal quotient of | #144718, #144719, #144746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6389);