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) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,18,23)(15,19,16,25)(17,21,27,22)(20,24,28,26);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,16)(14,19,23,25)(17,24,27,26)(20,21,28,22);(2,4)(13,16,18,15)(14,20,23,28)(17,24,27,26)(19,21,25,22);(6,10)(7,12)(14,21)(17,27)(19,28)(20,25)(22,23)(24,26)Polycyclic presentation:
2^8|a^2=eh,b^2=eh,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632124, #6632133, #6632139, … |
| maximal normal subgroup of | #6632124, #6632133, #6632139, … |
| maximal quotient of | #335125, #335127, #335133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23521);