Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26)(27,29)(28,35)(30,32)(31,33)(34,37)(36,39)(38,40);(1,2)(3,7)(4,8)(5,6)(10,15)(12,16)(13,22)(18,24)(25,27,31,28)(26,29,33,35)(30,36,38,34)(32,39,40,37);(2,6)(3,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28,31,27)(26,29,33,35)(30,36,38,34)(32,37,40,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30,31,38)(26,32,33,40)(27,34,28,36)(29,37,35,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=g,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58498, #8104369, #8104433, … |
| maximal normal subgroup of | C2×#58498, #8104369, #8104433, … |
| maximal quotient of | #585449, #586613, #586676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51486);