Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(17,18,21,20)(19,23,24,22);(1,3,5,8)(2,6,7,4)(10,14)(12,15);(9,10)(11,15)(12,16)(13,14)(17,19,21,24)(18,22,20,23);(1,4)(2,3)(5,6)(7,8)(9,11,13,16)(10,15,14,12);(1,3)(2,4)(5,8)(6,7)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=g,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40327, #7637518, #7638136, … |
| maximal normal subgroup of | C2×#40327, #7637518, #7638136, … |
| maximal quotient of | #521930, #522477, #522952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33315);