Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,18)(13,20)(14,19)(15,21)(17,25)(22,30)(23,35)(24,32)(26,34)(27,31)(28,33)(29,37)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,21,32)(12,22,26,38)(13,27,14,23)(17,29,33,40)(18,30,34,39)(19,35,20,31)(25,37,28,36);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,22)(13,25)(14,28)(15,26)(16,29)(18,19)(20,34)(21,33)(23,36)(24,38)(27,37)(30,35)(31,39)(32,40);(3,8)(6,7)(9,12)(10,18)(11,22)(13,28)(14,25)(15,26)(16,30)(17,19)(20,33)(21,34)(23,37)(24,38)(27,36)(29,35)(31,40)(32,39);(9,13,15,14)(10,19,21,20)(11,23,24,27)(12,25,26,28)(16,31,32,35)(17,18,33,34)(22,36,38,37)(29,39,40,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131102, #131104, #220610, … |
| maximal normal subgroup of | #131102, #131104, #220610, … |
| maximal quotient of | #6379327, #6379860, #6380086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54277);