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) = 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 | |
691 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)(3,7)(9,10,12,13)(11,14,16,15)(17,18,21,23)(19,24,22,20)(26,29)(27,31);(3,7)(6,8)(10,13)(14,15)(19,22)(20,24)(25,26,28,29)(27,30,31,32);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,14,13,15)(17,19,21,22)(18,20,23,24)(25,26)(27,32)(28,29)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=eg,b^2=efh,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×#17255, #845110, #845134, … |
| maximal normal subgroup of | C2×#17255, #845110, #845134, … |
| maximal quotient of | #156502, #156503, #156504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10243);