Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #486 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2099 |
| prime radical | C23 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,7,6,11,8,12)(3,13,15,26,17,22,16,9)(4,18,20,28,21,10,24,23)(14,19,29,31,27,32,30,25);(1,3,8,15,6,17,5,16)(2,9,12,13,11,26,7,22)(4,19,24,31,21,32,20,25)(10,27,28,30,18,14,23,29);(1,4,5,20,6,21,8,24)(2,10,7,23,11,18,12,28)(3,14,15,29,17,27,16,30)(9,25,13,19,26,31,22,32)Polycyclic presentation:
2^8|a^2=d,b^2=de,c^2=d,d^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,g^b=gh,f^c=fgh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126349, #126350, #126387, … |
| maximal normal subgroup of | #126349, #126350, #126387, … |
| maximal quotient of | #146283, #146284, #146285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6528);