Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 262144 |
| 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×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
982 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)(5,6,10,11)(7,8,9,12)(14,18)(16,24)(19,21)(22,28);(6,11)(7,9)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(1,2,3,4)(5,7,10,9)(6,12,11,8);(5,8)(6,7)(9,11)(10,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,9,10,7)(6,8,11,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97185, #97342, #130213, … |
| maximal normal subgroup of | #97185, #97342, #130213, … |
| maximal quotient of | #557080, #557388, #557704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39135);