Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #102 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,12,7,10)(14,19,20,23)(15,16)(17,25)(21,28,27,22)(24,26);(2,4)(6,10,12,11)(13,14,16,21,25,28,15,23,18,20,26,27,17,22,24,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7178, #126785, #126786, … |
| maximal normal subgroup of | C2×#7178, #126785, #126786, … |
| maximal quotient of | #92942, #92946, #92958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 166);