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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,5,7)(2,6,8,4)(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,30,32,31);(1,4,5,6)(2,7,8,3)(10,14)(13,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(2,8)(3,7)(11,16)(13,15)(18,23)(19,27)(24,32)(26,28)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=efh,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 | #6637350, #6637354, #6637378, … |
| maximal normal subgroup of | #6637350, #6637354, #6637378, … |
| maximal quotient of | #384276, #384609, #384683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25044);