Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1862 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,7,14,9,15,8,30)(3,16,20,46,21,47,31,34)(4,22,25,43,26,58,32,38)(5,13,27,42,28,44,6,12)(10,19,35,54,37,56,36,18)(11,23,39,60,41,29,40,24)(17,48,51,62,53,63,52,33)(45,55,61,59,64,49,57,50);(1,3,7,20,9,21,8,31)(2,10,14,35,15,37,30,36)(4,17,25,51,26,53,32,52)(5,19,27,54,28,56,6,18)(11,33,39,48,41,62,40,63)(12,34,13,16,42,46,44,47)(22,57,43,45,58,61,38,64)(23,59,60,49,29,50,24,55);(1,4)(2,11)(3,17)(5,24)(6,29)(7,25)(8,32)(9,26)(10,33)(12,22)(13,43)(14,39)(15,41)(16,45)(18,50)(19,55)(20,51)(21,53)(23,27)(28,60)(30,40)(31,52)(34,57)(35,48)(36,63)(37,62)(38,44)(42,58)(46,61)(47,64)(49,56)(54,59)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=egh,g^b=gh,d^c=dgh,e^c=egh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12947, #494317, #495719, … |
| maximal normal subgroup of | C2×#12947, #494317, #495719, … |
| maximal quotient of | #135888, #135889, #135926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5935);