Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C24×C4×#120 (order 2048) · Inn(G) = C24 · Out(G) = C22×C4×D4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
346 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)(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(2,4);(5,7,8,16,9,17,18,29,10,15,19,27,20,28,30,35)(6,11,12,22,13,23,24,33,14,21,25,31,26,32,34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36)Polycyclic presentation:
2^8|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #498112, #498113, #498114, … |
| maximal normal subgroup of | #498112, #498113, #498114, … |
| maximal quotient of | #483514, #483527, #483528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26424);