Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,13)(5,10)(6,15)(7,11)(8,12)(14,16)(17,18,21,26,22,24,20,25)(19,27,28,23,30,32,29,31);(1,3,7,4)(2,8,11,16)(5,12,6,14)(9,10,13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(1,4,7,3)(2,9,11,13)(5,12,6,14)(8,10,16,15)(18,24)(23,31)(25,26)(27,32);(1,5,7,6)(2,10,11,15)(3,12,4,14)(8,13,16,9)(17,20,22,21)(18,25,24,26)(19,28,30,29)(23,32,31,27)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17769, #6344072, #6344074, … |
| maximal normal subgroup of | C2×#17769, #6344072, #6344074, … |
| maximal quotient of | #160657, #160660, #160663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10757);