Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
765 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)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,29,28,31);(9,11,12,16)(10,13,14,15)(19,25)(21,23)(27,28)(29,31);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31);(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41703, #7639971, #7640031, … |
| maximal normal subgroup of | C2×#41703, #7639971, #7640031, … |
| maximal quotient of | #544807, #545251, #545785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34691);