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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | D16 |
| 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 | |
97 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)(7,12)(8,16)(9,19)(13,21)(14,30)(15,31)(17,33)(18,27)(20,32)(22,23)(24,29)(25,26)(28,36)(34,35);(5,6,12,23,8,14,25,31,19,29,33,36,32,34,18,21,10,13,27,35,20,28,17,24,9,15,26,30,16,22,7,11);(6,13)(11,21)(14,28)(15,29)(22,34)(23,35)(24,31)(30,36)Polycyclic presentation:
2^8|a^2=e,b^2=df,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #152000, #152421, #152423, … |
| maximal normal subgroup of | #152000, #152421, #152423, … |
| maximal quotient of | #151033, #151053, #151064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6622);