Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19);(1,3,4,2)(6,7)(11,18)(14,23)(17,24)(19,21);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(5,7,8,6)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #227082, #227084, #227087, … |
| maximal normal subgroup of | #227082, #227084, #227087, … |
| maximal quotient of | #229063, #229075, #229087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15992);