Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 98304 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,5,8)(10,15)(11,16)(18,21)(22,24);(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,12,16,13)(17,18,20,21)(19,22,23,24);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,15)(14,16);(1,4,6,2)(3,7,5,8)(9,12,14,13)(10,16,15,11);(1,5,6,3)(2,8,4,7)(9,13,14,12)(10,11,15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=h,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #36603⋊C3 |
| maximal subgroup of | C2×#36603, #7630321, #7630390, … |
| maximal normal subgroup of | C2×#36603, #7630321, #7630390, … |
| maximal quotient of | #519187, #519215, #519534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29591);