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×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,10,8)(7,11,9,12)(13,14)(15,19)(16,20)(17,18);(6,8)(7,9)(14,18)(16,19);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,17,20)(14,19,18,16);(5,8,10,6)(7,11,9,12)(13,16,17,19)(14,20,18,15);(5,9,10,7)(6,11,8,12)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=f,e^2=h,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 | #7001575, #7001576, #7001579, … |
| maximal normal subgroup of | #7001575, #7001576, #7001579, … |
| maximal quotient of | #535326, #535488, #535958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38088);