Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,23)(13,17)(14,27)(15,21)(16,25)(18,26)(20,35)(22,30)(24,32)(28,31)(29,39)(33,37)(34,38)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,40)(24,36)(25,37)(26,38)(27,34)(28,39)(31,33)(32,35);(9,12,13,24)(10,16,17,31)(11,20,21,36)(14,28,26,25)(15,29,19,33)(18,32,27,23)(22,39,38,37)(30,35,34,40);(3,8)(5,7)(11,21)(15,19)(20,36)(22,38)(29,33)(30,34)(35,40)(37,39);(10,17)(11,21)(16,31)(18,27)(20,36)(22,38)(23,32)(37,39)Polycyclic presentation:
2^8|a^2=fh,c^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #7054911, #7054921, #7054927, … |
| maximal normal subgroup of | #7054911, #7054921, #7054927, … |
| maximal quotient of | #6379081, #6379314, #6379717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54174);