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 | |
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)(7,17,19,12,21,28,20,27)(8,22,24,13,25,30,23,31)(18,26,33,32,35,29,34,36);(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,29)(14,28)(15,17)(16,27)(22,32)(23,34)(24,33)(25,35)(26,31)(30,36);(2,4)(5,8,11,25)(6,13,16,31)(7,18,21,35)(9,24,10,23)(12,26,27,29)(14,22,15,30)(17,32,28,36)(19,34,20,33)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,d^c=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11005, #357561, C4×#2009, … |
| maximal normal subgroup of | C2×#11005, #357561, C4×#2009, … |
| maximal quotient of | #111319, #111320, #111323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3993);