Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 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)(7,8)(9,10,13,15)(11,12,14,16)(18,19)(20,27)(24,26)(28,32);(9,11)(10,12)(13,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,15)(13,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,3,5,4)(2,7,6,8)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,4,5,3)(2,8,6,7)(9,12)(10,14)(11,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#38018, #7635218, #7635345, … |
| maximal normal subgroup of | C2×#38018, #7635218, #7635345, … |
| maximal quotient of | #521390, #522118, #522714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31006);