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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17,15,20,13,19)(11,16,21,32,24,29,23,31)(12,25,28,34,27,33,26,18)(22,36,37,39,38,40,35,30);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,22)(13,23)(14,21)(15,24)(17,32)(18,30)(19,31)(20,29)(25,36)(26,35)(27,38)(28,37)(33,40)(34,39);(1,4,5,8)(2,6,3,7)(10,17)(11,21)(13,14)(16,29)(18,33)(19,20)(22,35)(23,24)(25,34)(26,28)(30,39)(37,38);(2,3)(6,7)(9,12,15,27)(10,18,20,34)(11,22,24,38)(13,28,14,26)(16,30,29,39)(17,33,19,25)(21,35,23,37)(31,36,32,40)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=e,d^2=h,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 | #6669711, #6669713, #6669715, … |
| maximal normal subgroup of | #6669711, #6669713, #6669715, … |
| maximal quotient of | #344273, #344274, #344277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24175);