Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
445 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)(9,10,15,16,14,18,13,17)(11,12,20,23,21,24,19,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,19)(14,21)(15,20)(16,23)(17,22)(18,24);(10,16)(11,19)(13,15)(17,18)(20,21)(22,23);(2,6)(3,8)(9,12)(10,11)(13,23)(14,24)(15,22)(16,19)(17,20)(18,21)Polycyclic presentation:
2^8|a^2=fh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6667761, #6667765, #6667767, … |
| maximal normal subgroup of | #6667761, #6667765, #6667767, … |
| maximal quotient of | #342299, #342300, #342301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24042);