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 | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
973 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)(5,6,10,8)(7,13,15,11)(9,16,17,12)(14,18,20,19)(22,26)(24,32)(27,29)(30,36);(6,8)(11,13)(12,16)(18,19)(21,22,25,26)(23,27,31,29)(24,33,32,28)(30,35,36,34);(1,2,3,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(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=h,c^2=fg,d^2=fh,e^2=f,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 | #132432, #245968, #245994, … |
| maximal normal subgroup of | #132432, #245968, #245994, … |
| maximal quotient of | #560456, #560472, #561552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44199);