Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12)(7,8,11,10)(13,14,15,17)(16,18,20,19);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(14,17)(18,19);(2,4)(5,8,9,10)(6,11,12,7)(14,18)(16,20)(17,19);(6,12)(7,11)(14,19)(16,20)(17,18)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6633377, #6633379, #6633380, … |
| maximal normal subgroup of | #6633377, #6633379, #6633380, … |
| maximal quotient of | #338955, #338957, #338963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23815);