Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| 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×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1165 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)(5,6,9,8)(7,11,12,10)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28);(6,8)(10,11)(14,15)(16,23)(17,25)(26,27);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27);(5,8,9,6)(7,11,12,10)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=fh,c^2=f,d^2=f,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 | #130994, #245534, #245535, … |
| maximal normal subgroup of | #130994, #245534, #245535, … |
| maximal quotient of | #535314, #535486, #535943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36484);