Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
631 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)(7,16)(8,21,24,13)(9,25)(10,14)(11,15)(12,18)(17,29,34,33)(19,28)(20,32)(22,36,35,27)(23,30,26,31);(2,4)(5,7,11,20)(6,12,15,28)(8,22,24,35)(9,19,10,18)(13,29,21,33)(14,32,25,16)(17,26,34,23)(27,30,36,31);(5,8)(6,13)(7,17)(9,23)(10,26)(11,24)(12,27)(14,30)(15,21)(16,33)(18,22)(19,35)(20,34)(25,31)(28,36)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11031, #357586, C4×#2009, … |
| maximal normal subgroup of | C2×#11031, #357586, C4×#2009, … |
| maximal quotient of | #111291, #111292, #111484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4019);