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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C42 |
| Frattini, prime radical, cosocle | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6)(7,14,16,19)(8,12,17,11)(9,13)(10,15,18,20);(1,2,3,4)(5,7,8,16,9,15,17,20)(6,10,12,14,13,19,11,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=gh,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 | #94636, #94648, #94662, … |
| maximal normal subgroup of | #94636, #94648, #94662, … |
| maximal quotient of | #94105, #94157, #94181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 492);