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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
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)(16,33)(17,31)(19,36)(20,32)(21,29)(23,27)(24,30)(26,35)(28,34);(5,7,10,17)(6,11,13,27)(8,19,21,35)(9,23,25,18)(12,28,29,33)(14,31,22,15)(16,24,34,20)(26,32,36,30);(2,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,18)(16,26)(17,27)(19,28)(20,30)(21,29)(22,25)(23,31)(24,32)(33,35)(34,36);(5,8,10,21)(6,12,13,29)(7,16,17,34)(9,24,25,20)(11,26,27,36)(14,32,22,30)(15,33,31,28)(18,35,23,19);(6,13)(7,17)(12,29)(14,22)(16,34)(18,23)(19,35)(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,d^b=df,e^b=eh,f^b=fh,e^c=eh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250091, #250123, #250155, … |
| maximal normal subgroup of | #250091, #250123, #250155, … |
| maximal quotient of | #6359231, #6359238, #6359245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54085);