Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | |
1355 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)(7,16)(8,13)(9,14)(10,23)(11,15)(12,17)(18,33)(19,26)(20,29)(21,36)(22,31)(24,28)(25,32)(27,34)(30,35);(5,7,11,19)(6,12,15,28)(8,17,10,24)(9,21,22,34)(13,26,23,16)(14,30,31,33)(18,25,35,20)(27,29,36,32);(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)(12,28)(13,23)(14,31)(16,26)(27,36)(29,32)(30,33)Polycyclic presentation:
2^8|b^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,f^b=fh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130733, #130777, #497328, … |
| maximal normal subgroup of | #130733, #130777, #497328, … |
| maximal quotient of | #6358431, #6358440, #6358451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54042);