Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,9)(7,10,12,11)(13,14,16,18)(15,17,19,20)(21,22,25,27)(23,26,24,28);(6,9)(10,11)(15,19)(17,20)(21,23,25,24)(22,26,27,28);(5,7,8,12)(6,10,9,11)(13,15)(14,17)(16,19)(18,20)(21,24,25,23)(22,26,27,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,26)(25,27)Polycyclic presentation:
2^8|a^2=efg,b^2=f,c^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17182, #844967, #845007, … |
| maximal normal subgroup of | C2×#17182, #844967, #845007, … |
| maximal quotient of | #156477, #156501, #156513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10170);