Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
764 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)(7,8)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(1,2,6,3)(4,7,5,8)(9,10,12,13)(11,14,15,16)(19,26)(21,29)(23,25)(24,28);(1,3,6,2)(4,8,5,7)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,4,6,5)(2,7,3,8)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,5,6,4)(2,8,3,7)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132125, #253331, #253353, … |
| maximal normal subgroup of | #132125, #253331, #253353, … |
| maximal quotient of | #515444, #515512, #515624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28482);