Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,17,19,13)(12,20,23,16)(18,22,21,24)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,2,5,4)(3,6,7,8)(10,15)(13,17)(16,20)(22,24)(27,35)(29,39)(31,40)(33,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,21)(14,19)(15,17)(16,24)(18,23)(20,22)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(9,12,14,23)(10,16,15,20)(11,18,19,21)(13,22,17,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4,5,2)(3,8,7,6)(9,13)(10,11)(12,22)(14,17)(15,19)(16,18)(20,21)(23,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41664, #7640180, #7640239, … |
| maximal normal subgroup of | C2×#41664, #7640180, #7640239, … |
| maximal quotient of | #544793, #545249, #545855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34652);