Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12)(7,11,8,10)(13,14,15,18,16,19,20,17);(1,2)(3,4)(5,7,9,8)(6,10,12,11);(2,4)(14,17)(15,20)(18,19);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #268356, #268359, #268366, … |
| maximal normal subgroup of | #268356, #268359, #268366, … |
| maximal quotient of | #279243, #279249, #279255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17823);