Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72342683 (order 6144) · Inn(G) = C22 · Out(G) = C2×S4×#131 |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×#2258 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22 |
| commutator | C2 | C23×C16 |
| Frattini, cosocle | C16 | C24 |
| socle | C22 | C22×C16 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,6,11,7,12,19,29,8,13,20,30,22,32,41,51,9,14,21,31,23,33,42,52,24,34,43,53,44,54,59,64)(5,10,16,26,17,27,38,48,18,28,39,49,40,50,58,63,15,25,35,45,36,46,55,60,37,47,56,61,57,62,65,66);(3,5)(4,10)(6,16)(7,17)(8,18)(9,15)(11,26)(12,27)(13,28)(14,25)(19,38)(20,39)(21,35)(22,40)(23,36)(24,37)(29,48)(30,49)(31,45)(32,50)(33,46)(34,47)(41,58)(42,55)(43,56)(44,57)(51,63)(52,60)(53,61)(54,62)(59,65)(64,66);(5,15)(10,25)(16,35)(17,36)(18,37)(26,45)(27,46)(28,47)(38,55)(39,56)(40,57)(48,60)(49,61)(50,62)(58,65)(63,66)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26961);