Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,12,14,15)(18,22)(19,26)(20,29)(30,31);(2,5)(6,7)(9,10,13,14)(11,15,16,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|b^2=fg,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133441, #251956, #251958, … |
| maximal normal subgroup of | #133441, #251956, #251958, … |
| maximal quotient of | #576657, #576771, #577209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39445);