Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = D42 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | D4×C22⋊C4 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,17,19,25)(5,20)(6,10)(7,11)(8,15)(9,18,26,22)(13,28,32,23)(14,16)(21,24)(27,31,29,30);(1,3)(2,8)(4,13)(5,21)(6,15)(7,16)(9,23)(10,14)(11,24)(12,20)(17,27)(18,31)(19,32)(22,30)(25,29)(26,28);(1,4)(2,9)(3,13)(5,18)(6,22)(7,19)(8,23)(10,25)(11,26)(12,27)(14,29)(15,30)(16,32)(17,20)(21,31)(24,28);(1,2)(3,14)(4,9)(5,20)(6,10)(7,11)(8,21)(12,16)(13,29)(15,24)(17,18)(19,26)(22,25)(23,31)(27,32)(28,30)Polycyclic presentation:
2^8|a^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,f^c=fh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497430, #497431, #498126, … |
| maximal normal subgroup of | #497430, #497431, #498126, … |
| maximal quotient of | #220220, #220221, #220222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15449);