Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14,19,17)(15,22,16,20)(18,25,26,21)(23,27,24,28);(6,9)(10,12)(13,15,19,16)(14,20,17,22)(18,23,26,24)(21,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,17,22)(18,24,26,23)(21,27,25,28);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,17,19,14)(15,20,16,22)(18,25,26,21)(23,27,24,28);(13,18,19,26)(14,21,17,25)(15,23,16,24)(20,27,22,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=h,e^2=f,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 | #6398648, #6398652, #6398665, … |
| maximal normal subgroup of | #6398648, #6398652, #6398665, … |
| maximal quotient of | #513825, #513842, #513865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27451);