Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,12,18,27)(8,20)(9,16)(10,15)(11,14)(13,22)(17,31,33,34)(19,30,24,26)(21,28)(23,29)(25,35,36,32);(5,7,10,19,11,18,9,24)(6,12,16,26,14,27,15,30)(8,17,23,32,21,33,22,35)(13,25,28,31,29,36,20,34);(1,2,3,4)(5,8,9,22,11,21,10,23)(6,13,15,20,14,29,16,28)(7,17,24,35,18,33,19,32)(12,25,30,34,27,36,26,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10952, #354122, C4×#1902, … |
| maximal normal subgroup of | C2×#10952, #354122, C4×#1902, … |
| maximal quotient of | #110129, #110131, #110132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3940);