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 | |
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:
(2,3)(6,7)(11,15)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,15,16)(19,26)(21,29)(23,25)(24,28);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=g,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 | #130584, #130907, #253920, … |
| maximal normal subgroup of | #130584, #130907, #253920, … |
| maximal quotient of | #535408, #535559, #536126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38337);