Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1007 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,24)(13,19)(14,16)(15,28)(17,22)(18,26)(21,36)(23,32)(25,34)(27,33)(29,30)(31,37)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(10,16)(17,30)(18,24)(19,28)(20,32)(31,40)(33,34)(35,36);(3,8)(5,7)(9,11,14,23)(10,17,16,30)(12,21,26,38)(13,22,15,29)(18,31,24,40)(19,32,28,20)(25,37,27,39)(33,35,34,36);(2,6)(5,7)(9,12)(10,18)(11,21)(13,27)(14,26)(15,25)(16,24)(17,31)(19,34)(20,35)(22,39)(23,38)(28,33)(29,37)(30,40)(32,36);(9,13,14,15)(10,19,16,28)(11,22,23,29)(12,25,26,27)(17,32,30,20)(18,33,24,34)(21,37,38,39)(31,35,40,36)Polycyclic presentation:
2^8|b^2=fh,c^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133264, #152101, #249851, … |
| maximal normal subgroup of | #133264, #152101, #249851, … |
| maximal quotient of | #6374316, #6374471, #6374980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53819);