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)(3,16,20,34,21,47,31,43)(4,22,25,39,26,59,32,58)(5,13,28,44,29,45,6,12)(10,27,35,55,37,19,36,18)(11,38,40,60,42,23,41,24)(17,48,51,33,53,62,52,63)(46,56,64,54,61,49,57,50);(1,3,7,20,9,21,8,31)(2,10,14,35,15,37,30,36)(4,17,25,51,26,53,32,52)(5,27,28,55,29,19,6,18)(11,33,40,62,42,63,41,48)(12,43,13,16,44,34,45,47)(22,57,39,46,59,64,58,61)(23,54,24,49,38,50,60,56);(1,4)(2,11)(3,17)(5,24)(6,23)(7,25)(8,32)(9,26)(10,33)(12,22)(13,39)(14,40)(15,42)(16,46)(18,54)(19,56)(20,51)(21,53)(27,49)(28,38)(29,60)(30,41)(31,52)(34,64)(35,62)(36,48)(37,63)(43,57)(44,59)(45,58)(47,61)(50,55)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=eg,g^b=gh,d^c=dg,e^c=egh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12942, #494304, #495719, … |
| maximal normal subgroup of | C2×#12942, #494304, #495719, … |
| maximal quotient of | #135885, #135886, #135923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5930);