Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(9,10,12,14)(11,13,15,16)(18,22)(23,30)(24,27)(31,32);(2,4)(5,6,7,8)(11,15)(13,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,32,28,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(9,11)(10,13)(12,15)(14,16)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #112148, #112149, #112834, … |
| maximal normal subgroup of | #112148, #112149, #112834, … |
| maximal quotient of | #154051, #154053, #154073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7651);