Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,22)(15,20,25,21,16,26,27,28);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,17,27,19,16,18,25)(14,20,22,28,24,26,23,21);(6,10)(9,12)(13,16,19,15)(14,21,24,28)(17,27,18,25)(20,22,26,23);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^8|a^2=g,b^2=efh,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27786, #6457661, #6457668, … |
| maximal normal subgroup of | C2×#27786, #6457661, #6457668, … |
| maximal quotient of | #315327, #315328, #315331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20774);