Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 1048576 |
| 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 | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(6,9)(10,12)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(14,20)(18,21)(19,28)(26,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #105913, #106305, #106359, … |
| maximal normal subgroup of | #105913, #106305, #106359, … |
| maximal quotient of | #513389, #513403, C4×#2310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27047);