Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7414 |
| automorphism | Aut(G) = #72282281 (order 1024) · Inn(G) = #1345 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1345 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #423 | C22 |
| prime radical | #575 | C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
95 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)(3,11,13,23,15,27,14,30)(4,8,18,24)(5,20)(6,9)(7,12,21,28,22,29,26,16)(10,17)(19,32,31,25);(1,3)(2,7)(4,16)(5,13)(6,15)(8,23)(9,22)(10,26)(11,25)(12,19)(14,17)(18,28)(20,21)(24,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dg,e^c=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7415, #150231, #150451, … |
| maximal normal subgroup of | C2×#7415, #150231, #150451, … |
| maximal quotient of | #93813, #93821, #93845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 403);