Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,13,17)(11,14,19,23)(12,15,20,16)(18,21,24,22);(1,2,4,3)(5,6,7,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(6,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=egh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #258301, #258302, #258313, … |
| maximal normal subgroup of | #258301, #258302, #258313, … |
| maximal quotient of | #317894, #317895, #317896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21242);