Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,7,5)(3,8,6,4)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,22);(1,3,7,6)(2,8,5,4)(18,22)(19,24);(1,4)(2,6)(3,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23);(1,5,7,2)(3,8,6,4)(17,20)(18,19)(21,23)(22,24);(1,6,7,3)(2,8,5,4)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98080, #167979, #167980, … |
| maximal normal subgroup of | #98080, #167979, #167980, … |
| maximal quotient of | #553869, #553873, #554100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30208);