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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C43 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6)(3,4)(7,8)(9,10,12,14,13,16,11,15)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,2,4,7,5,8,3,6)(10,14)(11,12)(15,16)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,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 | #93813, #93821, C2×#7040, … |
| maximal normal subgroup of | #93813, #93821, C2×#7040, … |
| maximal quotient of | #92809, #92813, #93595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28);