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 | #2819 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(1,2,5,4)(3,6,7,8)(10,13)(14,15)(19,27)(21,31)(23,32)(25,28);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(9,11,12,16)(10,14,13,15)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,4,5,2)(3,8,7,6)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=g,d^2=f,e^2=g,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×#41735, #7639760, #7640241, … |
| maximal normal subgroup of | C2×#41735, #7639760, #7640241, … |
| maximal quotient of | #544835, #545274, #545795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34723);