Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1125 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)(6,9)(10,12)(14,17)(15,21)(20,24)(22,27);(13,14,18,17)(15,19,21,23)(16,24,25,20)(22,28,27,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,16,18,25)(14,20,17,24)(15,22,21,27)(19,26,23,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,23,21,19)(16,24,25,20)(22,28,27,26)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34192⋊C3 |
| maximal subgroup of | #106447, #106901, #107019, … |
| maximal normal subgroup of | #106447, #106901, #107019, … |
| maximal quotient of | #513563, #513689, #513716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27180);