Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,20,27,19,32,18,12)(8,13,22,30,23,29,25,28)(17,31,33,36,21,35,34,26);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,24,32)(8,21,22,34,23,17,25,33)(13,26,28,35,29,36,30,31);(2,4)(5,8)(6,13)(7,17)(9,25)(10,22)(11,23)(12,26)(14,30)(15,29)(16,31)(18,34)(19,21)(20,33)(24,28)(27,36)(32,35)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,f^b=fh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11137, #357831, #358505, … |
| maximal normal subgroup of | C2×#11137, #357831, #358505, … |
| maximal quotient of | #116991, #116992, #117027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4125);