Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| 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 | |
762 conjugacy classes of subgroups; only the 415 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 415 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,12)(11,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(2,5)(6,7)(9,10,13,12)(11,16,15,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,16,15,14)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=g,d^2=f,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 | #133177, #133179, #251734, … |
| maximal normal subgroup of | #133177, #133179, #251734, … |
| maximal quotient of | #553760, #553796, #553991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30383);