Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,11)(7,10,12,8)(13,14,17,19)(15,16,20,18);(7,12)(8,10)(13,15,17,20)(14,18,19,16);(1,2,3,4)(5,7,9,12)(6,8,11,10)(13,16)(14,15)(17,18)(19,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,20)(18,19);(5,8,9,10)(6,7,11,12)(13,16)(14,15)(17,18)(19,20)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51782, #7707778, #7707842, … |
| maximal normal subgroup of | C2×#51782, #7707778, #7707842, … |
| maximal quotient of | #560427, #560453, #561534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44770);