Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | |
734 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)(10,14)(15,16)(17,18,23,26)(19,27,20,24)(21,32,25,22)(28,29,30,31);(1,2)(3,6)(4,5)(7,8)(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20,23,19)(18,24,26,27)(21,28,25,30)(22,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97049, #131217, #131223, … |
| maximal normal subgroup of | #97049, #131217, #131223, … |
| maximal quotient of | #556511, #556513, #556524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30600);