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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,26,28,23)(20,30,25,22)(27,32,31,29);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,28)(18,23,24,26)(20,31,25,27)(22,29,30,32);(11,15)(13,16)(17,20,22,18,21,25,30,24)(19,27,29,26,28,31,32,23);(1,3,4,8)(2,6,5,7)(18,24)(19,28)(20,25)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29189, #6463864, #6463870, … |
| maximal normal subgroup of | C2×#29189, #6463864, #6463870, … |
| maximal quotient of | #362188, #363521, #365500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22177);