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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 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)(9,10,14,18,13,17,24,15)(11,19,20,23,16,12,21,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,20)(15,23)(17,19)(18,22)(21,24);(10,15)(11,20)(14,24)(16,21)(17,18)(22,23);(2,6)(3,8)(9,12)(10,16)(11,17)(13,19)(14,23)(15,21)(18,20)(22,24)Polycyclic presentation:
2^8|a^2=f,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 | #6667760, #6667764, #6667766, … |
| maximal normal subgroup of | #6667760, #6667764, #6667766, … |
| maximal quotient of | #342235, #342236, #342237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24040);