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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1004 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)(6,10)(11,14)(12,17)(18,20)(21,22)(23,26)(24,27)(25,30)(28,33)(29,32)(31,35)(34,36);(6,10)(7,13)(8,16)(18,20)(22,23)(25,33)(27,29)(31,36);(1,2)(3,4)(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^8|a^2=gh,c^2=f,d^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×#50868, #7706590, #7706655, … |
| maximal normal subgroup of | C2×#50868, #7706590, #7706655, … |
| maximal quotient of | #560555, #560556, #561650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43856);