Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28);(5,6)(7,11)(8,9)(10,12)(14,15)(16,23)(17,25)(26,27);(1,2)(3,4)(6,9)(10,11)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=gh,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46832, #7700704, #7700720, … |
| maximal normal subgroup of | C2×#46832, #7700704, #7700720, … |
| maximal quotient of | #576637, #576701, #576722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39820);