Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14)(7,8,12,11)(9,13,17,20)(15,16,19,18)(22,26)(24,32)(27,29)(30,36);(7,12)(8,11)(9,17)(13,20)(21,22,25,26)(23,27,31,29)(24,33,32,28)(30,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,12)(13,18)(16,20)(17,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,10,11)(6,12,14,7)(9,16,17,18)(13,19,20,15)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,10,17)(6,13,14,20)(7,15,12,19)(8,16,11,18)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=fh,b^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132062, #246952, #246956, … |
| maximal normal subgroup of | #132062, #246952, #246956, … |
| maximal quotient of | #560364, #560366, #561459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45044);