Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = C4×#7622880 (order 2048) · Inn(G) = #117 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1752 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,6,12,7,13,28,42,8,14,29,43,5,11,26,40)(3,15,18,44,17,33,47,41,19,39,27,55,25,9,30,32)(4,10,22,36,24,38,54,63,23,37,53,62,21,35,51,60)(16,34,45,58,20,48,49,64,46,59,56,61,50,31,52,57);(1,3)(2,9)(4,20,23,50)(5,25)(6,27)(7,17)(8,19)(10,34,37,59)(11,39)(12,41)(13,15)(14,33)(16,24,46,21)(18,29)(22,52,53,49)(26,47)(28,30)(31,38,48,35)(32,43)(36,61,62,58)(40,44)(42,55)(45,51,56,54)(57,60,64,63);(1,4,6,22,7,24,28,54,8,23,29,53,5,21,26,51)(2,10,12,36,13,38,42,63,14,37,43,62,11,35,40,60)(3,16,18,45,17,20,47,49,19,46,27,56,25,50,30,52)(9,31,32,57,15,34,44,58,33,48,41,64,39,59,55,61)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,c^b=cg,d^b=dh,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128481, #128482, #128483, … |
| maximal normal subgroup of | #128481, #128482, #128483, … |
| maximal quotient of | #128257, #128269, #128289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5605);