Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2535 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,16)(5,18)(8,11)(9,24)(10,21)(12,26)(13,14)(17,29)(19,23)(25,30)(27,28);(1,2,5,7,6,10,18,21)(3,11,14,23,15,19,13,8)(4,9,16,20,17,24,29,31)(12,25,27,32,28,30,26,22);(1,3)(2,8)(4,12)(5,13)(6,15)(7,19)(9,22)(10,23)(11,21)(14,18)(16,26)(17,28)(20,30)(24,32)(25,31)(27,29);(1,4,5,16,6,17,18,29)(2,9,7,20,10,24,21,31)(3,12,13,26,15,28,14,27)(8,22,19,30,23,32,11,25)Polycyclic presentation:
2^8|b^2=g,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28182, #6462911, #6462912, … |
| maximal normal subgroup of | C2×#28182, #6462911, #6462912, … |
| maximal quotient of | #316991, #316994, #316997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21170);