Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,14)(13,17)(15,20)(16,19)(21,22)(23,24);(10,14,17,16)(15,22,23,18);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,20)(21,23)(22,24);(9,11,19,24,13,21,12,20)(10,15,14,22,17,23,16,18)Polycyclic presentation:
2^8|a^2=f,b^2=egh,c^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144510, #144511, #144524, … |
| maximal normal subgroup of | #144510, #144511, #144524, … |
| maximal quotient of | #325801, #325802, #325807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21813);