Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,7)(3,8)(11,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|b^2=h,c^2=ef,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6636801, #6636813, #6636843, … |
| maximal normal subgroup of | #6636801, #6636813, #6636843, … |
| maximal quotient of | #384236, #384606, #384842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24987);