Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | #2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1001 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)(8,15)(12,17)(14,19)(18,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(6,10)(7,13)(12,17)(14,19)(22,26)(27,29)(28,33)(34,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^8|d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37121, #7634487, #7634501, … |
| maximal normal subgroup of | C2×#37121, #7634487, #7634501, … |
| maximal quotient of | #553743, #553748, #554005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30109);