Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 | |
613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14,17,15)(16,19,20,18)(22,24)(23,27);(6,9)(10,12)(14,15)(18,19)(21,22,25,24)(23,28,27,26);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23,25,27)(22,26,24,28);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^8|a^2=h,b^2=efg,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #224387, #224389, #224407, … |
| maximal normal subgroup of | #224387, #224389, #224407, … |
| maximal quotient of | #153762, #153763, #153847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6920);