Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(18,23,25,29)(24,30,32,31);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,27)(18,24,25,32)(20,28,21,26)(23,30,29,31);(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #220537, #220541, #220545, … |
| maximal normal subgroup of | #220537, #220541, #220545, … |
| maximal quotient of | #163739, #163740, #163748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10600);