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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
635 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)(5,6,10,14,11,16,9,15)(7,17,19,12,20,27,25,28)(8,13,23,30,24,31,22,32)(18,29,33,26,34,36,21,35);(5,7)(6,12)(8,21)(9,25)(10,19)(11,20)(13,29)(14,27)(15,17)(16,28)(18,23)(22,34)(24,33)(26,30)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,33)(20,34)(21,25)(27,36)(28,35)Polycyclic presentation:
2^8|a^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11018, #357575, #358190, … |
| maximal normal subgroup of | C2×#11018, #357575, #358190, … |
| maximal quotient of | #111455, #111502, #111569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4006);