Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,19,20,16);(2,3)(6,8)(10,11)(14,18)(16,19);(1,2,4,3)(13,15)(14,16)(17,20)(18,19);(5,7,9,12)(6,10,8,11)(13,15)(14,16)(17,20)(18,19);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39457, #7636616, #7636728, … |
| maximal normal subgroup of | C2×#39457, #7636616, #7636728, … |
| maximal quotient of | #521451, #522166, #522709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32445);