Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | #3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,21)(4,16)(8,20,23,31)(10,22,11,17)(13,27)(18,30)(24,32,25,28)(26,29);(1,2)(3,8)(4,17)(5,10)(6,11)(7,18)(9,19)(12,23)(13,28)(14,24)(15,25)(16,21)(20,29)(22,30)(26,32)(27,31);(1,3,5,14)(2,8,10,24)(4,13,16,27)(6,15,19,12)(7,20,22,32)(9,23,11,25)(17,28,21,31)(18,29,30,26);(3,12)(8,23)(13,26)(14,15)(20,31)(24,25)(27,29)(28,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh61 conjugacy classes — view the full character table.
| maximal subgroup of | #254277, #254280, #254283, … |
| maximal normal subgroup of | #254277, #254280, #254283, … |
| maximal quotient of | #160512, #160513, #160518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10603);