Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1012 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,18)(13,19)(14,28)(15,20)(17,25)(21,30)(22,31)(23,38)(24,32)(26,33)(27,34)(29,35)(36,39)(37,40);(2,5)(6,8)(9,11,15,24)(10,16,20,32)(12,21,26,36)(13,22,14,23)(17,29,34,40)(18,30,33,39)(19,31,28,38)(25,35,27,37);(9,12)(10,17)(11,21)(13,25)(14,27)(15,26)(16,29)(18,28)(19,33)(20,34)(22,35)(23,37)(24,36)(30,38)(31,39)(32,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,27)(14,25)(15,26)(16,30)(17,28)(19,34)(20,33)(22,37)(23,35)(24,36)(29,38)(31,40)(32,39);(9,13,15,14)(10,19,20,28)(11,22,24,23)(12,25,26,27)(16,31,32,38)(17,33,34,18)(21,35,36,37)(29,39,40,30)Polycyclic presentation:
2^8|b^2=h,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #6576740, #6576741, #6576752, … |
| maximal normal subgroup of | #6576740, #6576741, #6576752, … |
| maximal quotient of | #6371021, #6371818, #6371820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53574);