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) = 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 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,14,16,15)(18,22)(19,25)(20,29)(30,32);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,5,8)(2,6,4,7)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fh,d^2=f,e^2=g,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×#47560, #7700899, #7700955, … |
| maximal normal subgroup of | C2×#47560, #7700899, #7700955, … |
| maximal quotient of | #576840, #576872, #577089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40548);