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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,22)(13,26)(14,17)(16,24)(18,28)(19,35)(20,30)(21,38)(23,40)(25,34)(27,33)(29,36)(31,37)(32,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,30)(18,31)(19,25)(22,40)(24,39)(26,38)(27,36)(28,37)(29,33)(34,35);(9,12,14,25)(10,16,17,33)(11,19,20,23)(13,27,28,24)(15,29,30,32)(18,34,26,22)(21,39,37,36)(31,40,38,35);(2,5)(6,8)(10,17)(15,30)(16,33)(18,26)(22,34)(29,32)(31,38)(35,40);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(16,33)(18,26)(19,23)(21,37)(22,34)(36,39)Polycyclic presentation:
2^8|c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61549, #10576880, #10576939, … |
| maximal normal subgroup of | C2×#61549, #10576880, #10576939, … |
| maximal quotient of | #6379598, #6379608, #6380405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54537);