Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7410 |
| automorphism | Aut(G) = #72285068 (order 1024) · Inn(G) = #1344 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1344 |
| commutator | C22×C8 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,11,17,25)(3,8,13,22,14,23,30,24)(6,10,16,26,18,27,15,31)(7,19,20,12,21,28,32,29);(2,7,11,21)(3,12)(4,15)(6,18)(8,10,23,27)(9,22,25,24)(14,29)(16,17)(19,28)(20,31,32,26)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=f,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7411, #150207, #150218, … |
| maximal normal subgroup of | C2×#7411, #150207, #150218, … |
| maximal quotient of | #93791, #93805, #93833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 399);