Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,7)(3,5,8,4)(10,13)(11,15)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31);(1,3,6,8)(2,4,7,5)(9,10,12,13)(11,14,15,16)(17,19,21,27)(18,22,23,30)(20,25,28,26)(24,29,32,31);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,5,6,4)(2,3,7,8)(10,13)(14,16)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6647965, #6647977, #6647982, … |
| maximal normal subgroup of | #6647965, #6647977, #6647982, … |
| maximal quotient of | #384949, #384951, #385718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25631);