Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(2,3)(10,14)(15,17)(16,21)(22,23);(1,2,4,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(1,3,4,2)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^8|c^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #497296, #6533145, #6573727, … |
| maximal normal subgroup of | #497296, #6533145, #6573727, … |
| maximal quotient of | #521404, #522128, #522677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32321);