Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ 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) = 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)(17,18,20,24,21,23,29,31)(19,25,26,32,27,22,28,30);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16);(4,12)(7,14)(10,16)(13,15)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dgh55 conjugacy classes — view the full character table.
| maximal subgroup of | #138788, #138790, #138804, … |
| maximal normal subgroup of | #138788, #138790, #138804, … |
| maximal quotient of | #164342, #164345, #164349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10414);