Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | |
1516 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)(12,20)(15,21)(18,24)(22,23);(2,3)(9,10,13,16)(11,14,19,17)(12,21,20,15)(18,23,24,22);(5,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24);(1,3,4,2)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(9,12,13,20)(10,15,16,21)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #6996487, #6996488, #6996491, … |
| maximal normal subgroup of | #6996487, #6996488, #6996491, … |
| maximal quotient of | #535333, #535382, #535968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38229);