Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,18,10,8,11,13)(7,14,15,19,16,12,17,20);(2,4)(5,7,11,17,10,16,9,15)(6,12,13,19,8,14,18,20);(5,8,10,6)(7,12,16,14)(9,18,11,13)(15,19,17,20);(2,4)(5,9,10,11)(6,13,8,18)(7,15,16,17)(12,19,14,20)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25147, #6438879, #6438883, … |
| maximal normal subgroup of | C2×#25147, #6438879, #6438883, … |
| maximal quotient of | #281568, #281569, #281572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18135);