Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | |
1337 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)(5,6,11,16)(7,17,21,32)(8,13,23,30)(9,15,10,14)(12,25,28,20)(18,33,36,27)(19,34,35,26)(22,31,24,29);(5,7,11,21)(6,12,16,28)(8,18,23,36)(9,20,10,25)(13,26,30,34)(14,17,15,32)(19,24,35,22)(27,31,33,29);(2,4)(5,8)(6,13)(7,18)(9,24)(10,22)(11,23)(12,26)(14,31)(15,29)(16,30)(17,33)(19,25)(20,35)(21,36)(27,32)(28,34);(5,8)(6,13)(7,19)(9,22)(10,24)(11,23)(12,27)(14,29)(15,31)(16,30)(17,34)(18,25)(20,36)(21,35)(26,32)(28,33);(5,9,11,10)(6,14,16,15)(7,20,21,25)(8,22,23,24)(12,17,28,32)(13,29,30,31)(18,19,36,35)(26,27,34,33)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,f^b=fh,e^c=eh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497341, #497342, #497349, … |
| maximal normal subgroup of | #497341, #497342, #497349, … |
| maximal quotient of | #6359038, #6359044, #6359045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54071);