Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,12)(17,18,22,23)(19,24,27,30)(20,25,29,32)(21,28,31,26);(2,3)(4,8)(9,10,14,13)(11,16,12,15)(18,23)(19,27)(20,29)(21,31);(2,3)(6,7)(9,11,14,12)(10,15,13,16)(17,19)(18,24)(20,26)(21,25)(22,27)(23,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,13,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,32)(24,31)(26,27);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,12,15)(17,21,22,31)(18,26,23,28)(19,25,27,32)(20,30,29,24)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133002, #487106, #487234, … |
| maximal normal subgroup of | #133002, #487106, #487234, … |
| maximal quotient of | #585438, #586216, #587557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50313);