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 | |
1361 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)(14,23)(16,29)(17,26)(19,25)(20,33)(21,27)(24,31)(28,34)(30,36)(32,35);(5,7,10,19)(6,11,13,26)(8,16,21,32)(9,17,23,18)(12,24,27,36)(14,25,22,15)(20,30,34,31)(28,35,33,29);(2,4)(5,8,10,21)(6,12,13,27)(7,16,19,32)(9,20,23,34)(11,24,26,36)(14,28,22,33)(15,29,25,35)(17,30,18,31);(7,17)(9,23)(11,25)(14,22)(15,26)(16,30)(18,19)(20,34)(24,35)(28,33)(29,36)(31,32);(6,13)(11,26)(12,27)(14,22)(15,25)(24,36)(28,33)(29,35)Polycyclic presentation:
2^8|b^2=h,c^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 | #497326, #6573733, #6792407, … |
| maximal normal subgroup of | #497326, #6573733, #6792407, … |
| maximal quotient of | #6358429, #6358438, #6358447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54040);