Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2661 | C2 |
| 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:
(1,2,3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,19)(14,17)(18,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,33)(34,36);(6,10)(11,17)(12,15)(18,19)(21,23,26,28)(22,27,30,24)(25,33,34,31)(29,32,36,35);(1,2)(3,4)(5,7,9,14)(6,11,10,17)(8,13,16,20)(12,18,15,19)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,34)(35,36);(5,6)(7,11)(8,12)(9,10)(13,18)(14,17)(15,16)(19,20)(21,24)(22,28)(23,30)(25,32)(26,27)(29,33)(31,36)(34,35);(5,8)(6,12)(7,13)(9,16)(10,15)(11,18)(14,20)(17,19)(21,25,26,34)(22,29,30,36)(23,31,28,33)(24,32,27,35)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=f,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 | C2×#51976, #7708017, #7708019, … |
| maximal normal subgroup of | C2×#51976, #7708017, #7708019, … |
| maximal quotient of | #560522, #560543, #561621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44964);