Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C23 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,9,13,22,8,11,15,27,23,25,32,36,24)(7,16,19,17,20,12,28,29,21,30,26,35,18,31,34,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,20)(10,26)(11,21)(13,30)(14,33)(15,31)(16,32)(18,25)(19,27)(22,34)(23,29)(24,35)(28,36);(5,8,22,32,25,14,10,15,11,24,36,13,9,23,27,6)(7,17,34,16,18,33,26,31,21,35,28,30,20,29,19,12)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^b=dh,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #128694, #128695, #129164, … |
| maximal normal subgroup of | #128694, #128695, #129164, … |
| maximal quotient of | #129044, #129060, #129072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5653);