Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#107 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
277 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)(5,6,9,14,11,17,23,16)(7,18,21,33,19,32,35,28)(8,15,10,13,24,30,25,31)(12,22,27,34,26,36,29,20);(2,4)(5,7,10,22,9,21,24,34,11,19,25,36,23,35,8,20)(6,12,15,28,16,29,31,32,17,26,30,33,14,27,13,18);(7,19)(12,26)(18,32)(20,34)(21,35)(22,36)(27,29)(28,33)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #146345, #146447, #146454, … |
| maximal normal subgroup of | #146345, #146447, #146454, … |
| maximal quotient of | #126649, #126687, #126702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5547);