Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,14,12,16,11,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(3,8)(5,7)(9,11,12,13)(10,14,16,15)(17,19,22,29)(18,23,25,26)(20,28,21,27)(24,32,30,31);(1,3,4,8)(2,5,6,7)(9,11,12,13)(10,15,16,14)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(2,6)(5,7)(9,11,12,13)(10,15,16,14)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cefh,d^b=deh52 conjugacy classes — view the full character table.
| maximal subgroup of | #100102, #100103, #100104, … |
| maximal normal subgroup of | #100102, #100103, #100104, … |
| maximal quotient of | #357309, #357384, #357410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16354);