Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,17)(11,19,21,15)(12,23,25,39)(14,27,24,18)(16,30,31,26)(20,34,36,38)(22,29,35,33)(28,37,32,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,19)(18,29)(23,38)(24,35)(25,36)(26,37)(27,33)(30,40)(31,32)(34,39);(9,12,24,30,13,25,14,26)(10,16,27,39,17,31,18,23)(11,20,35,40,21,36,22,37)(15,28,33,34,19,32,29,38);(2,5)(6,8)(11,21)(15,19)(20,36)(22,35)(28,32)(29,33)(34,38)(37,40);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(16,31)(18,27)(20,36)(22,35)(23,39)(37,40)Polycyclic presentation:
2^8|a^2=h,c^2=gh,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131669, #131673, #263741, … |
| maximal normal subgroup of | #131669, #131673, #263741, … |
| maximal quotient of | #6386889, #6389510, #6389541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55425);