Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,4,6)(10,14)(11,17)(12,20)(15,23)(16,18)(21,24);(1,3,5,4)(2,6,7,8)(10,15)(12,20)(14,23)(16,21)(18,24)(19,22);(1,4,5,3)(2,6,7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| Fitting subgroup of | #13464⋊C3 |
| maximal subgroup of | #151404, #151405, #151408, … |
| maximal normal subgroup of | #151404, #151405, #151408, … |
| maximal quotient of | #145777, #145793, #145804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6452);