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 | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 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)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(18,23)(19,27)(24,31)(26,28)Polycyclic presentation:
2^8|a^2=fh,b^2=efh,c^2=eh,d^2=eh,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 | #6635489, #6635499, #6635500, … |
| maximal normal subgroup of | #6635489, #6635499, #6635500, … |
| maximal quotient of | #383298, #384110, #384202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24949);