Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,25,20,24,29,23)(19,26,28,31,27,22,30,32);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(17,19)(18,22)(20,27)(21,28)(23,31)(24,26)(25,32)(29,30);(9,11,12,16)(10,14,13,15)(18,23)(21,29)(22,31)(24,25)(26,32)(28,30);(2,6)(5,7)(10,13)(14,15)(18,24)(19,27)(23,25)(28,30)Polycyclic presentation:
2^8|a^2=fg,b^2=egh,c^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29342, #6461415, #6461421, … |
| maximal normal subgroup of | C2×#29342, #6461415, #6461421, … |
| maximal quotient of | #362806, #364359, #368479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22330);