Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = C2×#72268043 (order 2048) · Inn(G) = C24 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #121 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(2,5)(6,8)(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^8|a^2=f,d^2=e,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=de,c^b=ceh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #135869, #135875, #136747, … |
| maximal normal subgroup of | #135869, #135875, #136747, … |
| maximal quotient of | #493297, #493300, #493307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26497);