Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
785 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(2,5)(6,7)(9,11,14,12)(10,15,16,13)(18,22)(19,25)(20,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,16,13)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,6)(4,5)(7,8)(9,13,14,15)(10,11,16,12)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=h,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 | #7010373, #7010374, #7010385, … |
| maximal normal subgroup of | #7010373, #7010374, #7010385, … |
| maximal quotient of | #535550, #535551, #536191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38213);