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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1347 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)(5,6,11,15)(7,16,19,26)(8,13,10,23)(9,14,22,31)(12,24,28,17)(18,33,35,30)(20,29,25,32)(21,36,34,27);(5,7)(6,12)(8,17)(9,21)(10,24)(11,19)(13,26)(14,30)(15,28)(16,23)(18,20)(22,34)(25,35)(27,32)(29,36)(31,33);(2,4)(5,8,11,10)(6,13,15,23)(7,17,19,24)(9,20,22,25)(12,26,28,16)(14,29,31,32)(18,34,35,21)(27,30,36,33);(5,9,11,22)(6,14,15,31)(7,18,19,35)(8,20,10,25)(12,27,28,36)(13,29,23,32)(16,33,26,30)(17,34,24,21);(6,15)(7,19)(13,23)(14,31)(17,24)(18,35)(21,34)(29,32)Polycyclic presentation:
2^8|a^2=h,c^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,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6993881, #6993886, #6993894, … |
| maximal normal subgroup of | #6993881, #6993886, #6993894, … |
| maximal quotient of | #6358849, #6358857, #6358865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54066);