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 | #3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,22)(3,13,15,9,16,26,23,27)(4,17,19,28,5,20,21,10)(14,25,31,32,18,24,30,29);(1,3,7,15,8,16,6,23)(2,9,11,26,12,27,22,13)(4,18,19,30,5,14,21,31)(10,25,17,32,28,24,20,29);(1,4,8,5)(2,10,12,28)(3,14,16,18)(6,19,7,21)(9,24,27,25)(11,20,22,17)(13,29,26,32)(15,30,23,31);(1,5,8,4)(2,10,12,28)(3,14,16,18)(6,19,7,21)(9,25,27,24)(11,17,22,20)(13,29,26,32)(15,31,23,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^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×#25199, #6438902, #6438903, … |
| maximal normal subgroup of | C2×#25199, #6438902, #6438903, … |
| maximal quotient of | #283584, #283585, #283588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18187);