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 | #3508 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,27)(20,25)(21,31)(22,26)(23,28)(24,29)(30,32);(2,6)(4,11)(7,12)(10,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,25)(26,32)(27,31);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,25)(26,27)(31,32);(3,9)(7,12)(10,15)(14,16)(17,20,22,21)(18,25,26,31)(19,28,29,30)(23,24,32,27)Polycyclic presentation:
2^8|a^2=f,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×#17766, #6344072, #6344073, … |
| maximal normal subgroup of | C2×#17766, #6344072, #6344073, … |
| maximal quotient of | #160654, #160658, #160664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10754);