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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(15,19)(17,20);(2,3)(6,9)(7,12)(13,14)(15,20)(16,18)(17,19);(5,6,8,9)(7,10,12,11)(13,14,16,18)(15,17,19,20);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^8|a^2=fg,c^2=h,d^2=fg,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×#44001, #7668255, #7668713, … |
| maximal normal subgroup of | C2×#44001, #7668255, #7668713, … |
| maximal quotient of | #535390, #535613, #536049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36989);