Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1149 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)(5,6,9,10)(7,8,11,12)(14,18)(15,21)(16,24)(26,28);(6,10)(7,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(1,2,3,4)(5,7)(6,8)(9,11)(10,12);(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97269, #131296, #147786, … |
| maximal normal subgroup of | #97269, #131296, #147786, … |
| maximal quotient of | #553756, #554019, #554305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30374);