Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
561 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)(11,14)(12,13)(15,16)(18,21)(19,23)(25,26,28,30)(27,29,32,31);(3,7)(6,8)(10,13)(11,15)(17,18,20,21)(19,22,23,24)(25,26)(27,29)(28,30)(31,32);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^8|a^2=eg,b^2=h,c^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17241, #845095, #845134, … |
| maximal normal subgroup of | C2×#17241, #845095, #845134, … |
| maximal quotient of | #156514, #156521, #156529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10229);