Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,11)(8,9)(12,15)(13,14)(16,17)(19,20)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,33,36,32);(2,4)(5,6,10,14)(7,15,18,19)(8,13,9,11)(12,16,20,17)(21,23,26,34)(22,27,30,36)(24,33,25,32)(28,35,29,31);(5,7,10,18)(6,12,14,20)(8,17,9,16)(11,15,13,19)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #10319⋊C3 |
| maximal subgroup of | C2×#10319, #233365, #233400, … |
| maximal normal subgroup of | C2×#10319, #233365, #233400, … |
| maximal quotient of | #102847, #102850, #102872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3307);