Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 4096 |
| 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 | #3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,27)(7,17,19,13,20,29,26,12)(8,21,23,33,25,14,24,32)(18,34,22,28,35,31,36,30);(1,2)(3,4)(5,7,11,20)(6,12,16,13)(8,22,25,36)(9,26,10,19)(14,31,21,34)(15,29,27,17)(18,24,35,23)(28,33,30,32);(2,4)(5,7)(6,13)(8,18)(9,19)(10,26)(11,20)(12,16)(14,30)(15,29)(17,27)(21,28)(22,23)(24,36)(25,35)(31,33)(32,34);(5,8,11,25)(6,14,16,21)(7,18,20,35)(9,24,10,23)(12,28,13,30)(15,33,27,32)(17,34,29,31)(19,36,26,22)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6667753, #6667757, #6667765, … |
| maximal normal subgroup of | #6667753, #6667757, #6667765, … |
| maximal quotient of | #333137, #333140, #333152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23469);