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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
750 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)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,25)(20,28)(21,27)(22,23)(24,29)(26,31)(30,32);(1,3,5,8)(2,6,4,7)(10,13)(14,15)(18,23)(20,29)(21,25)(26,30);(3,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,30)(22,27)(23,25)(24,32)(26,29)(28,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,32)(27,30);(1,4,5,2)(3,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,21)(18,19)(20,31)(22,25)(23,27)(24,26)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,e^2=fg,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×#54466, #8102558, #8102626, … |
| maximal normal subgroup of | C2×#54466, #8102558, #8102626, … |
| maximal quotient of | #585301, #586553, #587108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47454);