Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C22×C22⋊C4 · Out(G) = C25 |
| 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 | #3520 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
498 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)(3,12)(4,15)(8,23,25,31)(10,22,11,16)(13,27)(14,30)(17,19)(20,24,32,26)(28,29);(1,2,5,10)(3,8,13,24)(4,16,15,21)(6,11,18,9)(7,19,22,17)(12,26,27,25)(14,23,29,32)(20,30,31,28);(1,3)(2,8)(4,14)(5,13)(6,12)(7,20)(9,25)(10,24)(11,26)(15,29)(16,23)(17,28)(18,27)(19,30)(21,32)(22,31);(3,13)(8,24)(12,27)(14,29)(20,31)(23,32)(25,26)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17598, #6343388, #6343389, … |
| maximal normal subgroup of | C2×#17598, #6343388, #6343389, … |
| maximal quotient of | #160631, #160636, #160643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10586);