Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #381 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1434 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
314 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)(3,11,13,7)(4,15)(5,10)(6,9)(8,16)(12,19,25,23)(14,24,27,20)(17,28)(18,22)(21,29)(26,30,32,31);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,19,11,23)(10,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10831, #354856, #355478, … |
| maximal normal subgroup of | C2×#10831, #354856, #355478, … |
| maximal quotient of | #108556, #108557, #108558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3819);