Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 65536 |
| 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 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,13)(14,16)(20,27)(24,29)(26,31)(30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,29)(26,31);(1,3,5,2)(4,7,8,6)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6447357, #6447358, #6447361, … |
| maximal normal subgroup of | #6447357, #6447358, #6447361, … |
| maximal quotient of | #516441, #516476, #516599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28171);