Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,18)(15,21)(20,25)(22,27);(6,7)(10,11)(14,18)(19,24)(20,25)(26,28);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,7,9,6)(8,11,12,10)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(5,8,9,12)(6,10,7,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107678, #190303, #197284, … |
| maximal normal subgroup of | #107678, #190303, #197284, … |
| maximal quotient of | #513788, #513798, #513799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27385);