Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #392 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1444 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,15,27,14,8)(4,9,20,24,19,25,18,31)(13,28,30,22,17,26,29,32);(1,3,5,14,7,15,6,16)(2,8,10,12,11,23,21,27)(4,17,18,30,19,13,20,29)(9,22,24,26,25,32,31,28);(1,4)(2,9)(3,13,15,17)(5,18)(6,20)(7,19)(8,22,23,32)(10,24)(11,25)(12,26,27,28)(14,29,16,30)(21,31)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11334, #359071, #360762, … |
| maximal normal subgroup of | C2×#11334, #359071, #360762, … |
| maximal quotient of | #121365, #121379, #121380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4322);