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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(13,16)(17,18,22,23)(19,25,28,24)(20,29,30,21)(26,27,32,31);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,23)(21,29)(24,25)(27,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,28)(18,24,23,25)(20,26,30,32)(21,27,29,31);(1,3,4,7)(2,6,5,8)(17,20)(18,21)(19,26)(22,30)(23,29)(24,27)(25,31)(28,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,20)(19,27)(22,29)(23,30)(24,26)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=h,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×#38395, #7635841, #7636310, … |
| maximal normal subgroup of | C2×#38395, #7635841, #7636310, … |
| maximal quotient of | #521760, #522355, #522972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31383);