Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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)(5,6,9,8)(7,11,12,10)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28);(1,2,3,4)(5,6,9,8)(7,10,12,11)(14,18)(15,22)(16,25)(26,27);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28);(1,2)(3,4)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=gh,e^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44677, #7668786, #7669110, … |
| maximal normal subgroup of | C2×#44677, #7668786, #7669110, … |
| maximal quotient of | #535370, #535657, #536038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37665);