Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,8)(11,15)(13,16)(17,18)(19,23)(20,22)(21,28)(24,29)(25,27)(26,31)(30,32);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(18,20)(23,25)(24,28)(30,31);(1,2,5,6)(3,4,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,3,5,7)(2,4,6,8)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,4)(2,7)(3,6)(5,8)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|b^2=gh,c^2=fg,d^2=fh,e^2=f,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 | C2×#37204, #7634562, #7634565, … |
| maximal normal subgroup of | C2×#37204, #7634562, #7634565, … |
| maximal quotient of | #553776, #553807, #554002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30192);