Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,7)(8,10,12,11)(14,17)(18,19)(21,22)(23,26)(24,27)(25,28);(6,7)(10,11)(13,14,16,17)(15,18,20,19)(21,23)(22,25)(24,28)(26,27);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,23)(22,26)(24,28)(25,27);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,27)(26,28)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14349⋊C3 |
| maximal subgroup of | #99163, #101085, #101089, … |
| maximal normal subgroup of | #99163, #101085, #101089, … |
| maximal quotient of | #153934, #153952, #154101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7337);