Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,25)(16,22)(17,32)(18,34)(19,24)(20,27)(21,30)(23,33)(26,39)(28,35)(29,41)(31,38)(36,43)(37,42)(40,44);(2,4)(6,9)(10,11)(13,15,16,26,18,28,30,42,19,29,31,43,33,44,17,27)(14,20,32,40,23,36,38,41,24,37,21,35,34,39,22,25);(5,7,8,12)(6,10,9,11)(13,16,18,30,19,31,33,17)(14,21,23,22,24,32,34,38)(15,26,28,42,29,43,44,27)(20,35,36,25,37,40,39,41)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #131919, #131949, #138571, … |
| maximal normal subgroup of | #131919, #131949, #138571, … |
| maximal quotient of | #141883, #141910, #142000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6269);