Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,11)(7,15)(8,17)(9,21)(12,24)(13,27)(14,28)(16,20)(18,34)(19,31)(22,33)(23,25)(26,35)(29,30)(32,36);(5,6)(7,16)(8,20)(9,23)(10,12)(11,15)(13,17)(14,21)(18,35)(19,32)(22,36)(24,31)(25,34)(26,33)(27,29)(28,30);(6,12)(11,24)(13,26)(14,23)(16,32)(20,36)(25,28)(27,35)Polycyclic presentation:
2^8|a^2=e,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=df,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #151972, #152165, #152174, … |
| maximal normal subgroup of | #151972, #152165, #152174, … |
| maximal quotient of | #150057, #150120, #150135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6604);