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)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(18,22,24,27)(23,29,30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,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 | #220534, #220538, #220542, … |
| maximal normal subgroup of | #220534, #220538, #220542, … |
| maximal quotient of | #163732, #163740, #163743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10593);