Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,14)(7,17,19,12,21,26,20,27)(8,13,23,30,22,28,24,29)(18,31,33,25,32,36,34,35);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,26,16,27,14,17)(8,18,24,34,22,32,23,33)(13,25,30,36,28,35,29,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,25)(14,29)(15,30)(16,28)(17,31)(19,33)(20,34)(21,32)(26,36)(27,35);(8,22)(13,28)(18,32)(23,24)(25,35)(29,30)(31,36)(33,34)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,e^a=eh,g^a=gh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #20351⋊C3 |
| maximal subgroup of | #138252, #138253, #356501, … |
| maximal normal subgroup of | #138252, #138253, #356501, … |
| maximal quotient of | #180912, #180918, #180925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13339);