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 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,20)(12,24)(13,18)(14,19)(15,28)(16,23)(17,25)(21,36)(22,32)(26,33)(27,31)(29,34)(30,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,27)(14,23)(15,22)(17,30)(18,32)(19,20)(24,40)(25,39)(26,38)(28,31)(29,37)(33,36)(34,35);(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,28)(27,39)(31,40)(32,36);(2,5)(6,8)(9,13,14,15)(10,18,19,28)(11,22,23,27)(12,25,26,29)(16,31,20,32)(17,33,34,24)(21,37,38,39)(30,40,35,36);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,34)(18,28)(21,38)(22,27)(24,33)(37,39)Polycyclic presentation:
2^8|d^2=h,e^2=fh,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 | C2×#61528, #10576867, #10576931, … |
| maximal normal subgroup of | C2×#61528, #10576867, #10576931, … |
| maximal quotient of | #6379589, #6379595, #6380434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54516);