Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
455 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)(7,10)(8,9)(11,12)(13,14)(15,24)(16,20)(17,22)(18,27)(19,23)(21,26)(25,28);(6,9)(10,12)(13,15,18,16,19,26,17,25)(14,20,22,24,23,28,27,21);(5,7,8,11)(6,10,9,12)(13,16,18,26,19,25,17,15)(14,21,22,20,23,24,27,28);(13,17,19,18)(14,22,23,27)(15,25,26,16)(20,24,28,21)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #110003, #116541, #116542, … |
| maximal normal subgroup of | #110003, #116541, #116542, … |
| maximal quotient of | #191166, #191174, #191182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13750);