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 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,24,23,20);(3,5)(7,8)(9,10,14,12)(11,15,13,16)(18,22)(19,23);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,12,16)(17,19,21,23)(18,20,22,24);(1,4,6,2)(3,7,5,8)(9,12,14,10)(11,15,13,16)(17,20)(18,23)(19,22)(21,24);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^8|a^2=h,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 | #131615, #131619, #484780, … |
| maximal normal subgroup of | #131615, #131619, #484780, … |
| maximal quotient of | #544513, #545107, #545375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35970);