Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
972 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)(7,11,13,15)(9,12,17,16)(14,18,19,20)(21,22,26,27)(23,24,29,32)(25,28,33,34)(30,31,35,36);(7,13)(9,17)(11,15)(12,16)(22,27)(23,29)(28,34)(30,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,23)(22,24)(25,30)(26,29)(27,32)(28,31)(33,35)(34,36);(5,8,10,6)(7,11,13,15)(9,16,17,12)(14,18,19,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,33)(22,28,27,34)(23,30,29,35)(24,31,32,36)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42176, #7640679, #7640690, … |
| maximal normal subgroup of | C2×#42176, #7640679, #7640690, … |
| maximal quotient of | #544535, #545090, #545531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35164);