Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(2,5)(3,7)(9,11,12,16)(10,13,15,14)(18,22)(23,25)(24,29)(30,31);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|b^2=g,c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6545478, #6545480, #6545498, … |
| maximal normal subgroup of | #6545478, #6545480, #6545498, … |
| maximal quotient of | #521708, #522324, #522828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31106);