Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,20)(9,12)(10,15)(11,16)(14,22)(17,25)(18,32)(19,27)(21,26)(23,30)(24,29)(28,31)(33,36)(34,35);(6,12,16,15)(7,17,19,24)(8,21)(13,27)(14,20,30,26)(18,33,34,31)(22,23)(25,29);(2,4)(5,7)(6,13)(8,18)(9,17)(10,24)(11,19)(12,25)(14,28)(15,29)(16,27)(20,32)(21,34)(22,31)(23,33)(26,35)(30,36);(5,8,9,22,11,21,10,23)(6,14,15,20,16,30,12,26)(7,18,17,31,19,34,24,33)(13,28,29,32,27,36,25,35)Polycyclic presentation:
2^8|a^2=f,b^2=eg,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28506, #6465678, #6465679, … |
| maximal normal subgroup of | C2×#28506, #6465678, #6465679, … |
| maximal quotient of | #321935, #321937, #321938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21494);