Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 262144 |
| 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 | |
1557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24);(2,3)(10,13)(11,17)(12,20)(23,24);(5,6,7,8)(9,11,14,17)(10,15,13,19)(12,21,20,18)(16,24,22,23);(1,3,4,2)(9,12,14,20)(10,16,13,22)(11,18,17,21)(15,23,19,24);(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)Polycyclic presentation:
2^8|a^2=fh,c^2=gh,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130751, #130754, #245747, … |
| maximal normal subgroup of | #130751, #130754, #245747, … |
| maximal quotient of | #535411, #535412, #536069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38375);