Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(14,16)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,29,32,31);(2,5)(6,8)(9,10,13,12)(11,14,15,16)(18,23)(19,26)(25,28)(27,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=fg,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 | #132010, #253376, #253378, … |
| maximal normal subgroup of | #132010, #253376, #253378, … |
| maximal quotient of | #553844, #554085, #554223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29822);