Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,12,7)(8,16,18,20)(9,10,14,11)(13,19,15,17);(1,2)(3,4)(5,7,10,14,12,6,11,9)(8,13,17,16,18,15,19,20);(2,4)(5,8)(6,13)(7,15)(9,16)(10,17)(11,19)(12,18)(14,20);(5,9,10,7,12,14,11,6)(8,16,17,15,18,20,19,13)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24284, #6442583, #6442584, … |
| maximal normal subgroup of | C2×#24284, #6442583, #6442584, … |
| maximal quotient of | #262788, #262789, #262791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17272);