Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(10,12)(13,14,17,16)(15,19,20,18)(22,23)(26,27);(15,20)(18,19)(21,22,25,23)(24,26,28,27);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,18,16,19)(21,23,25,22)(24,26,28,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #191088, #191089, #191092, … |
| maximal normal subgroup of | #191088, #191089, #191092, … |
| maximal quotient of | #153937, #153940, #153943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7340);