Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | #3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,17)(11,15,20,31)(12,22,25,34)(13,26,28,18)(16,27,33,24)(19,35,23,40)(21,38,37,30)(29,39,32,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,31)(18,30)(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)(15,29,31,32)(18,34,26,22)(21,39,37,36)(30,40,38,35);(3,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,29)(17,31)(18,30)(22,35)(23,25)(24,36)(26,38)(27,39)(28,37)(32,33)(34,40);(10,17)(11,20)(16,33)(18,26)(19,23)(21,37)(22,34)(36,39)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,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #250525, #250545, #250557, … |
| maximal normal subgroup of | #250525, #250545, #250557, … |
| maximal quotient of | #6379101, #6379328, #6379724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54209);