Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
391 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)(7,9)(8,10)(11,12)(13,14,16,20,17,21,24,18)(15,19,22,27,23,26,28,25);(5,7,8,11)(6,9,10,12)(14,18)(15,22)(16,24)(20,21)(23,28)(25,27);(6,10)(9,12)(15,23)(19,26)(22,28)(25,27);(13,15,16,22,17,23,24,28)(14,19,20,27,21,26,18,25)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22621, #6379847, #6379854, … |
| maximal normal subgroup of | C2×#22621, #6379847, #6379854, … |
| maximal quotient of | #225238, #225255, #225259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15609);