Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C22×C42 | C22 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25,26,22)(20,24)(21,23)(27,31,32,29)(28,30);(2,6,8,7)(9,11)(10,13,14,15)(12,16)(17,19,20,27,21,26,28,32)(18,22,24,29,23,25,30,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=fh,e^2=h,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93825, #93835, C2×#7046, … |
| maximal normal subgroup of | #93825, #93835, C2×#7046, … |
| maximal quotient of | #92811, #92812, #93637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34);