Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,12,14)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,13)(15,16)(18,19)(20,27)(24,26)(28,32);(11,12)(14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,14,15,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=fh,c^2=g,d^2=gh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58396, #8104193, #8104424, … |
| maximal normal subgroup of | C2×#58396, #8104193, #8104424, … |
| maximal quotient of | #585694, #585972, #586166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51384);