Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,8,7,4)(9,10)(11,16)(12,14)(13,15)(17,18,21,19)(20,23,24,22);(2,6)(3,7)(10,12)(15,16)(18,19)(22,23);(9,11,14,13)(10,15,12,16)(17,19,21,18)(20,23,24,22);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,16,13,15)(17,20,21,24)(18,22,19,23);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,16,12,15)Polycyclic presentation:
2^8|a^2=fg,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37102, #7634293, #7634383, … |
| maximal normal subgroup of | C2×#37102, #7634293, #7634383, … |
| maximal quotient of | #553952, #553964, #554152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30090);