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 ◁ #525 ◁ 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)(7,15)(8,12)(9,22)(10,13)(11,18)(14,25)(16,31)(17,33)(19,36)(20,32)(21,29)(23,26)(24,30)(27,35)(28,34);(5,7,10,16)(6,11,13,26)(8,19,21,35)(9,23,25,18)(12,28,29,33)(14,31,22,15)(17,24,34,20)(27,32,36,30);(2,4)(7,16)(11,26)(15,31)(17,34)(18,23)(19,35)(27,36)(28,33);(5,8,10,21)(6,12,13,29)(7,17,16,34)(9,24,25,20)(11,27,26,36)(14,32,22,30)(15,33,31,28)(18,35,23,19);(6,13)(11,26)(12,29)(14,22)(15,31)(27,36)(28,33)(30,32)Polycyclic presentation:
2^8|b^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497332, #6573738, #6792447, … |
| maximal normal subgroup of | #497332, #6573738, #6792447, … |
| maximal quotient of | #6358589, #6358596, #6358605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54049);