Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2424 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,21,9)(3,12,15,23,16,25,14,7)(4,17,19,24,20,26,18,8)(13,22,28,31,29,32,30,27);(1,3,5,15,6,16,21,14)(2,7,10,12,11,23,9,25)(4,13,19,28,20,29,18,30)(8,22,17,31,24,32,26,27);(1,4,6,20)(2,8,11,24)(3,13,16,29)(5,18,21,19)(7,22,23,32)(9,17,10,26)(12,27,25,31)(14,28,15,30);(1,2)(3,7)(4,8)(5,10)(6,11)(9,21)(12,15)(13,22)(14,25)(16,23)(17,19)(18,26)(20,24)(27,30)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25198, #6438902, #6438904, … |
| maximal normal subgroup of | C2×#25198, #6438902, #6438904, … |
| maximal quotient of | #283583, #283586, #283587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18186);