Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,8,11,10)(13,14)(15,23)(16,21)(17,19)(18,22)(20,25)(24,27)(26,28);(5,7,9,11)(6,10,12,8)(14,19)(15,24)(16,18)(20,27)(23,28)(25,26);(2,4)(5,8,9,10)(6,11,12,7)(13,15,17,25)(14,20,19,23)(16,24,18,26)(21,28,22,27);(2,4)(6,12)(7,11)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,28,23,27)Polycyclic presentation:
2^8|a^2=eh,b^2=eh,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632484, #6632492, #6632497, … |
| maximal normal subgroup of | #6632484, #6632492, #6632497, … |
| maximal quotient of | #338841, #338842, #338845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23774);