Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
724 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)(10,13)(14,16)(17,18,22,25)(19,20,27,23)(21,29,31,24)(26,32,30,28);(2,5)(3,7)(17,19,22,27)(18,23,25,20)(21,30,31,26)(24,28,29,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(17,21,22,31)(18,24,25,29)(19,26,27,30)(20,28,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #7221563, #7221567, #7221627, … |
| maximal normal subgroup of | #7221563, #7221567, #7221627, … |
| maximal quotient of | #557270, #557343, #557476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39039);