Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | Aut(G) = #72287397 (order 4096) · Inn(G) = #353 · Out(G) = D42 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C42 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1393 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,11,17,26)(3,12,14,28,13,23,29,25)(5,8,19,24,20,10,18,27)(7,16,21,30,22,31,32,15);(1,3)(2,7)(4,16,17,31)(5,15)(6,13)(8,23)(9,25,26,28)(10,12)(11,22)(14,19,29,18)(20,30)(21,24,32,27)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,d^2=h,f^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,c^b=ch,d^b=dfgh,d^c=dgh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94635, #94647, #94660, … |
| maximal normal subgroup of | #94635, #94647, #94660, … |
| maximal quotient of | #94106, #94159, #94182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 495);