Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,12,13)(11,14,16,15)(17,18,21,20)(19,23,24,22)(25,26,28,30)(27,31,32,29);(10,13)(14,15)(19,24)(22,23)(25,26,28,30)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,23,24,22)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16729, #611474, #611505, … |
| maximal normal subgroup of | C2×#16729, #611474, #611505, … |
| maximal quotient of | #155693, #155697, #155780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9717);