Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #469 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2065 |
| 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 | |
152 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 152 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:
(1,2,7,33,8,15,32,14)(3,16,48,37,22,45,21,56)(4,23,34,44,29,58,28,41)(5,13,30,11,31,46,6,12)(9,35,38,18,39,20,54,19)(10,27,42,25,43,60,47,26)(17,36,51,64,52,63,57,62)(24,59,55,40,53,61,50,49);(1,3,5,18,7,21,30,35,8,22,31,19,32,48,6,20)(2,9,11,16,14,38,13,37,15,39,12,45,33,54,46,56)(4,24,25,52,28,55,27,57,29,53,26,17,34,50,60,51)(10,40,23,36,42,59,44,62,43,49,58,63,47,61,41,64);(1,4)(2,10)(3,17)(5,27)(6,26)(7,34)(8,29)(9,36)(11,44)(12,41)(13,23)(14,47)(15,43)(16,49)(18,24)(19,53)(20,55)(21,57)(22,52)(25,30)(28,32)(31,60)(33,42)(35,50)(37,59)(38,64)(39,63)(40,45)(46,58)(48,51)(54,62)(56,61)Polycyclic presentation:
2^8|a^2=de,b^2=d,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,e^b=eh,d^c=dg,e^c=egh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13468, #499308, #499508, … |
| maximal normal subgroup of | C2×#13468, #499308, #499508, … |
| maximal quotient of | #145908, #145918, #145951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6456);