Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
939 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)(5,6,10,13)(7,11,14,9)(8,12,18,20)(15,19,16,17)(21,22,26,24)(23,29,31,27)(25,32,34,28)(30,35,33,36);(7,14)(9,11)(15,16)(17,19)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,13)(10,14)(12,17)(15,18)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,36)(33,34);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(21,24,26,22)(23,27,31,29)(25,32,34,28)(30,35,33,36);(5,9,10,11)(6,7,13,14)(8,17,18,19)(12,15,20,16)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47822, #7701902, #7701904, … |
| maximal normal subgroup of | C2×#47822, #7701902, #7701904, … |
| maximal quotient of | #576643, #576703, #576712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40810);