Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
977 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)(5,6,9,10)(7,13,15,11)(8,16,17,12)(14,18,20,19)(21,22,26,29)(23,24,30,27)(25,28,33,35)(31,32,36,34);(6,10)(8,17)(11,13)(14,20)(23,30)(24,27)(25,33)(28,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,24,26,27)(22,23,29,30)(25,32,33,34)(28,31,35,36);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^8|a^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98468, #246769, #246770, … |
| maximal normal subgroup of | #98468, #246769, #246770, … |
| maximal quotient of | #557045, #557052, #557365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39120);