Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14)(15,18)(16,19)(17,22)(20,25)(21,24)(23,27)(26,28);(5,6)(7,9)(8,11)(10,12)(14,15)(17,25)(19,21)(23,28);(1,2)(3,4)(6,7)(10,11)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(5,7,9,6)(8,11,12,10)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,8)(6,10)(7,11)(9,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|c^2=f,d^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50636, #7706334, #7706406, … |
| maximal normal subgroup of | C2×#50636, #7706334, #7706406, … |
| maximal quotient of | #560391, #560392, #561491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43624);