Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1070 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)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,35)(21,30)(22,31)(23,32)(26,33)(28,34)(29,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,29)(18,30)(19,32)(24,35)(25,36)(26,38)(27,31)(28,37)(33,39)(34,40);(3,8)(5,7)(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,33)(19,34)(21,36)(22,37)(23,38)(24,27)(30,39)(31,35)(32,40);(2,6)(5,7)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,32,31)(17,33,34,24)(20,36,38,37)(29,39,40,35);(9,13,15,14)(10,18,19,27)(11,22,23,21)(12,25,26,28)(16,31,32,30)(17,33,34,24)(20,37,38,36)(29,35,40,39)Polycyclic presentation:
2^8|d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6667752, #6667753, #6667764, … |
| maximal normal subgroup of | #6667752, #6667753, #6667764, … |
| maximal quotient of | #6374173, #6374177, #6374889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53754);