Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C22×D16 | 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 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,23)(10,25)(11,16)(14,19)(15,20)(17,26)(18,32)(21,29)(22,28)(24,36)(27,33)(30,34)(31,35);(6,12)(7,10)(8,19)(9,20)(13,29)(14,15)(16,28)(17,33)(18,24)(21,34)(22,35)(23,27)(25,26)(30,36)(31,32);(5,7,11,10)(6,13,16,25)(8,17,22,21)(9,18,24,20)(12,26,28,29)(14,30,31,27)(15,23,32,36)(19,34,35,33)Polycyclic presentation:
2^8|a^2=e,c^2=h,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 | #151969, #152169, #152176, … |
| maximal normal subgroup of | #151969, #152169, #152176, … |
| maximal quotient of | #150058, #150121, #150136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6605);