Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,15)(11,12,16,14)(18,22)(23,25)(24,29)(30,31);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(18,22)(20,28)(23,25)(26,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131572, #131576, #484787, … |
| maximal normal subgroup of | #131572, #131576, #484787, … |
| maximal quotient of | #544564, #545073, #545440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35971);