Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #418 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1804 |
| 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 | |
189 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,13,8,11,7,30)(3,14,17,35,18,9,29,34)(4,19,22,51,20,52,31,41)(5,23,26,39,28,55,27,12)(10,25,36,53,38,59,37,21)(15,43,46,64,44,60,47,62)(16,48,50,40,24,56,58,33)(32,49,61,45,42,57,63,54);(1,3,7,17,8,18,6,29)(2,9,30,34,11,14,13,35)(4,15,31,46,20,44,22,47)(5,24,27,58,28,16,26,50)(10,32,37,61,38,42,36,63)(12,40,55,56,39,33,23,48)(19,43,41,64,52,60,51,62)(21,54,59,49,53,45,25,57);(1,4)(2,10)(3,15)(5,25)(6,22)(7,31)(8,20)(9,32)(11,38)(12,41)(13,36)(14,42)(16,49)(17,46)(18,44)(19,23)(21,27)(24,57)(26,53)(28,59)(29,47)(30,37)(33,62)(34,61)(35,63)(39,51)(40,64)(43,48)(45,50)(52,55)(54,58)(56,60)Polycyclic presentation:
2^8|a^2=f,b^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12830, #493800, #494050, … |
| maximal normal subgroup of | C2×#12830, #493800, #494050, … |
| maximal quotient of | #134392, #134417, #134477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5818);