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 | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,25)(20,22)(21,28)(23,27)(24,29)(26,30)(31,32);(10,13)(14,15)(18,20)(23,25)(24,28)(30,31);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=g,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 | #6471910, #6471914, #6471919, … |
| maximal normal subgroup of | #6471910, #6471914, #6471919, … |
| maximal quotient of | #553818, #553837, #554060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30390);