Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,12,17,29)(8,13,21,31)(9,22,25,15)(11,23,27,18)(16,28,34,30)(19,36,35,26)(20,33,24,32);(5,7)(6,11)(8,19)(9,23)(10,17)(12,22)(13,30)(14,27)(15,29)(16,20)(18,25)(21,35)(24,34)(26,32)(28,31)(33,36);(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,22)(13,28)(14,29)(15,27)(19,20)(21,34)(23,25)(24,35)(26,32)(30,31)(33,36);(5,8,10,21)(6,13,14,31)(7,16,17,34)(9,24,25,20)(11,26,27,36)(12,28,29,30)(15,33,22,32)(18,35,23,19);(6,14)(7,17)(13,31)(15,22)(16,34)(18,23)(19,35)(32,33)Polycyclic presentation:
2^8|a^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,e^b=eh,f^b=fh,e^c=eh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #7142811, #7142813, #7142824, … |
| maximal normal subgroup of | #7142811, #7142813, #7142824, … |
| maximal quotient of | #6359230, #6359237, #6359239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54084);