Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #72287153 (order 4096) · Inn(G) = #354 · Out(G) = D42 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×C4⋊Q8 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,8,13,10,15,22,32)(7,16,18,33,17,28,34,30)(9,14,23,31,24,12,21,29)(11,25,26,19,27,20,36,35);(1,2,3,4)(5,7,9,19,10,17,24,35)(6,11,14,16,15,27,12,28)(8,20,21,18,22,25,23,34)(13,30,29,36,32,33,31,26)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,c^b=ch,d^b=dfg,d^c=dgh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94637, #94649, #94664, … |
| maximal normal subgroup of | #94637, #94649, #94664, … |
| maximal quotient of | #94105, #94156, #94183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 493);