Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | Aut(G) = #72281382 (order 2048) · Inn(G) = H(3,ℤ/4) · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | #405 | C22 |
| socle | C2 | #1290 |
| prime radical | C23 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,27,40,8,13,28,41,30,36,6,29)(3,14,16,43,18,45,17,51,19,46,25,44,52,62,53,63)(4,20,21,54,23,39,22,38,24,56,26,55,57,10,35,37)(9,31,32,48,33,59,60,49,34,50,61,64,42,47,58,15);(1,3,6,17,30,52,28,16,8,19,27,53,7,18,5,25)(2,9,11,32,12,33,40,60,13,34,41,61,36,42,29,58)(4,15,22,47,57,64,21,50,24,49,35,59,23,48,26,31)(10,14,38,44,20,45,55,63,39,46,37,43,56,62,54,51)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=f,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93810, #93811, #93814, … |
| maximal normal subgroup of | #93810, #93811, #93814, … |
| maximal quotient of | #93327, #93355, #93571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 227);