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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,20)(16,21)(17,24)(18,19)(22,27)(23,26)(25,28);(2,3)(5,6,9,8)(7,10,11,12)(14,19)(15,23)(16,25)(17,27);(5,7,9,11)(6,10,8,12)(13,15,18,23)(14,20,19,26)(16,22,25,24)(17,21,27,28);(1,3,4,2)(13,16,18,25)(14,21,19,28)(15,24,23,22)(17,26,27,20);(5,8,9,6)(7,12,11,10)(13,17)(14,22)(15,21)(16,26)(18,27)(19,24)(20,25)(23,28)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=gh,d^2=f,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×#44665, #7668826, #7669109, … |
| maximal normal subgroup of | C2×#44665, #7668826, #7669109, … |
| maximal quotient of | #535428, #535754, #536082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37653);