Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
897 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)(3,7)(4,6)(5,8)(9,10,13,18)(11,17,21,15)(12,23,25,39)(14,27,24,19)(16,30,32,26)(20,34,36,38)(22,29,35,33)(28,37,31,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,21)(10,15,18,17)(12,20,25,36)(14,22,24,35)(16,28,32,31)(19,29,27,33)(23,38,39,34)(26,37,30,40);(9,12,24,30,13,25,14,26)(10,16,27,39,18,32,19,23)(11,20,35,40,21,36,22,37)(15,28,33,34,17,31,29,38);(3,8)(5,7)(9,11)(10,17)(12,20)(13,21)(14,22)(15,18)(16,31)(19,33)(23,34)(24,35)(25,36)(26,37)(27,29)(28,32)(30,40)(38,39);(2,6)(5,7)(10,18)(11,21)(12,25)(19,27)(22,35)(26,30)(28,31)(34,38)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=gh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62352, #10577651, #10577666, … |
| maximal normal subgroup of | C2×#62352, #10577651, #10577666, … |
| maximal quotient of | #6386790, #6389223, #6389224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55340);