Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 196608 |
| 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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(15,16)(18,21)(22,24);(1,2,6,5)(3,8,4,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14);(1,4,6,3)(2,7,5,8)(9,11,13,12)(10,15,14,16);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,16,14,15)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36084⋊C3 |
| maximal subgroup of | #132278, #253633, #253635, … |
| maximal normal subgroup of | #132278, #253633, #253635, … |
| maximal quotient of | #517611, #517620, #517624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29072);