Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C22×#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,19,27)(12,23,25,31)(15,20)(16,21)(17,22)(18,26,29,32)(28,30);(1,3)(2,7,10,19)(4,12)(5,13)(6,14)(8,18,20,29)(9,11,22,24)(15,25)(16,26)(17,27)(21,23,30,31)(28,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,18)(9,21)(10,20)(11,23)(13,26)(14,25)(17,28)(19,29)(22,30)(24,31)(27,32);(4,15)(8,20)(12,25)(16,28)(18,29)(21,30)(23,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=be,g^a=gh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #20338⋊C3 |
| maximal subgroup of | #138266, #138267, #356478, … |
| maximal normal subgroup of | #138266, #138267, #356478, … |
| maximal quotient of | #180860, #180864, #180868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13326);