Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16)(8,20,22,29)(9,23)(10,14)(11,15)(12,18)(13,21,30,25)(17,32,35,26)(19,27)(24,28)(31,34,36,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,27)(15,28)(16,23)(20,31)(21,34)(22,35)(25,33)(29,36)(30,32);(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,23)(24,34)(27,36)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10779, #353454, #354320, … |
| maximal normal subgroup of | C2×#10779, #353454, #354320, … |
| maximal quotient of | #105862, #105863, #105868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3767);