Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(17,18,21,22)(19,24,23,20);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(18,22)(19,23);(3,8)(6,7)(10,11)(14,15)(17,19,21,23)(18,20,22,24);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(17,20)(18,23)(19,22)(21,24);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131617, #131621, #484781, … |
| maximal normal subgroup of | #131617, #131621, #484781, … |
| maximal quotient of | #544512, #545108, #545374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35963);