Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16)(7,17,19,28)(8,22,24,30)(9,15,10,14)(12,21,27,20)(13,25,29,23)(18,32,33,26)(31,35,36,34);(5,7,9,20,11,19,10,21)(6,12,14,17,16,27,15,28)(8,18,23,34,24,33,25,35)(13,26,22,31,29,32,30,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(1,3,4,2)(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=gh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #495738, #495900, #499515, … |
| maximal normal subgroup of | #495738, #495900, #499515, … |
| maximal quotient of | #211043, #211046, #211055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14544);