Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | |
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)(9,10,14,20)(11,18,24,16)(12,25,27,35)(13,21,15,19)(17,29,33,26)(22,36,38,40)(23,31,28,34)(30,39,32,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,28)(14,24)(15,23)(17,30)(18,20)(19,34)(21,31)(25,40)(26,39)(27,38)(29,37)(32,33)(35,36);(9,12,14,27)(10,17,20,33)(11,22,24,38)(13,26,15,29)(16,30,18,32)(19,25,21,35)(23,37,28,39)(31,36,34,40);(3,8)(5,7)(9,11)(10,18)(12,22)(13,28)(14,24)(15,23)(16,20)(17,32)(19,31)(21,34)(25,36)(26,39)(27,38)(29,37)(30,33)(35,40);(2,6)(5,7)(9,13,14,15)(10,19,20,21)(11,23,24,28)(12,26,27,29)(16,31,18,34)(17,25,33,35)(22,37,38,39)(30,36,32,40)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6363777, #6363781, #6363819, … |
| maximal normal subgroup of | #6363777, #6363781, #6363819, … |
| maximal quotient of | #6386069, #6387740, #6387746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54681);