Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1221 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(7,12)(9,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(5,7,8,12)(6,9,10,11)(15,22)(18,21)(23,28)(25,27);(6,10)(9,11)(14,19)(15,22)(20,26)(23,28);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107352, #107464, #107738, … |
| maximal normal subgroup of | #107352, #107464, #107738, … |
| maximal quotient of | #513760, #513764, #513768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27334);