Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | Aut(G) = #72286925 (order 4096) · Inn(G) = #1408 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1408 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | #494 | C22 |
| prime radical | C22 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,8,13,3,14)(4,10,19,26,22,29,18,9)(6,12,16,27,7,24,17,30)(15,31,20,28,23,25,21,32);(1,3,8,5)(2,9,13,26)(4,18,22,19)(6,17,7,16)(10,27,29,30)(11,28,14,32)(12,31,24,25)(15,21,23,20)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fg,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=def,e^b=egh,f^b=fh,d^c=dfg,e^d=egh,f^d=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #94822, #94825, #94826, … |
| maximal normal subgroup of | #94822, #94825, #94826, … |
| maximal quotient of | #94568, #94584, #94645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 520);