Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,8,7,4)(11,15)(13,16)(17,18,22,25)(19,23,26,21)(20,24,30,32)(27,31,28,29);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(19,26)(21,23)(27,28)(29,31);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(1,4)(2,7)(3,6)(5,8)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,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 | #6678396, #6678401, #6678461, … |
| maximal normal subgroup of | #6678396, #6678401, #6678461, … |
| maximal quotient of | #544424, #545010, #545492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35644);