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 | #2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 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)(9,10,13,11)(12,15,16,14)(20,28)(23,30)(26,32)(29,31);(10,11)(14,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,3,4,7)(2,6,5,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fg,d^2=fh,e^2=g,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 | #129630, #499275, #508186, … |
| maximal normal subgroup of | #129630, #499275, #508186, … |
| maximal quotient of | #553943, #554147, #554624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30215);