Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10)(3,7,14,21)(4,8,15,23)(6,11,17,9)(12,26,27,22)(13,19,29,20)(16,25,30,24)(18,31,32,28);(1,3,5,14)(2,7,10,21)(4,12,15,27)(6,13,17,29)(8,22,23,26)(9,20,11,19)(16,28,30,31)(18,25,32,24);(1,4,6,16)(2,8,11,25)(3,12,13,28)(5,15,17,30)(7,18,19,26)(9,24,10,23)(14,27,29,31)(20,22,21,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eh,d^c=dg34 conjugacy classes — view the full character table.
| Fitting subgroup of | #10339⋊C3 |
| maximal subgroup of | #151343, C2×#10339, #233540, … |
| maximal normal subgroup of | #151343, C2×#10339, #233540, … |
| maximal quotient of | #102887, #102888, #103189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3327);