Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 131072 |
| 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 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,14,13)(11,16,12,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(2,6)(4,7)(9,11,14,12)(10,15,13,16)(18,22)(19,26)(20,29)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #132744, #132800, #487463, … |
| maximal normal subgroup of | #132744, #132800, #487463, … |
| maximal quotient of | #535550, #536192, #537093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38214);