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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,14)(15,16)(19,26)(23,31)(27,32)(28,30);(1,3,5,8)(2,6,7,4)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,27,26,32)(20,24,22,29)(23,30,31,28);(10,14)(13,16)(17,19,21,26)(18,23,25,31)(20,28,22,30)(24,32,29,27);(1,4,5,6)(2,3,7,8)(10,14)(13,16)(17,20,21,22)(18,24,25,29)(19,28,26,30)(23,32,31,27)Polycyclic presentation:
2^8|a^2=e,b^2=ef,c^2=h,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 | #6632844, #6632846, #6632850, … |
| maximal normal subgroup of | #6632844, #6632846, #6632850, … |
| maximal quotient of | #383630, #384054, #384886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24525);