Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,23)(7,16,18,12,19,27,24,28)(8,13,22,32,21,30,25,31)(17,33,35,26,34,29,20,36);(5,7,10,18,11,19,9,24)(6,12,14,27,15,28,23,16)(8,20,22,17,21,35,25,34)(13,29,32,36,30,33,31,26);(2,4)(8,21)(13,30)(17,34)(20,35)(22,25)(26,36)(29,33)(31,32);(5,8)(6,13)(7,17)(9,25)(10,22)(11,21)(12,26)(14,32)(15,30)(16,33)(18,35)(19,34)(20,24)(23,31)(27,29)(28,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,d^c=dh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22887, #6381455, #6381456, … |
| maximal normal subgroup of | C2×#22887, #6381455, #6381456, … |
| maximal quotient of | #227160, #227161, #227162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15875);