Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | #2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,15,16,13)(18,22)(20,27)(23,28)(26,31);(9,11,12,16)(10,13,14,15)(18,22)(19,25)(24,29)(26,31);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6482741, #6482745, #6482748, … |
| maximal normal subgroup of | #6482741, #6482745, #6482748, … |
| maximal quotient of | #553811, #553828, #554058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30405);