Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(15,21)(18,22)(23,27)(26,28);(6,10)(7,11)(13,14,17,20)(15,22,21,18)(16,24,25,19)(23,26,27,28);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(5,7,9,11)(6,8,10,12)(13,15,17,21)(14,18,20,22)(16,23,25,27)(19,26,24,28);(5,8)(6,11)(7,10)(9,12)(13,16,17,25)(14,19,20,24)(15,23,21,27)(18,26,22,28)Polycyclic presentation:
2^8|b^2=h,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133340, #499871, #500197, … |
| maximal normal subgroup of | #133340, #499871, #500197, … |
| maximal quotient of | #557053, #557370, #557664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39298);