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 | #3012 | 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:
(5,6,9,15,11,14,10,24)(7,16,19,28,20,27,18,12)(8,21,22,13,23,30,25,29)(17,32,34,36,35,26,33,31);(1,2)(3,4)(5,7,11,20)(6,12,14,28)(8,17,23,35)(9,18,10,19)(13,26,29,32)(15,27,24,16)(21,31,30,36)(22,33,25,34);(2,4)(5,8,11,23)(6,13,14,29)(7,17,20,35)(9,22,10,25)(12,26,28,32)(15,30,24,21)(16,31,27,36)(18,33,19,34);(6,14)(7,18)(9,10)(12,16)(13,29)(17,33)(19,20)(22,25)(26,31)(27,28)(32,36)(34,35)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^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, #6667755, #6667765, … |
| maximal normal subgroup of | #6667753, #6667755, #6667765, … |
| maximal quotient of | #333128, #333133, #333143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23470);