Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,22)(13,20)(14,18)(15,19)(17,23)(21,24);(1,3,4,8)(2,5,6,7)(9,11,13,21,15,17,14,22)(10,16,20,24,19,23,18,12);(9,12,13,23,15,24,14,16)(10,17,18,22,19,11,20,21);(2,6)(5,7)Polycyclic presentation:
2^8|b^2=egh,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28593, #6459592, #6459597, … |
| maximal normal subgroup of | C2×#28593, #6459592, #6459597, … |
| maximal quotient of | #323861, #323862, #323863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21581);