Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 32768 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
497 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)(3,10,13,7)(4,8)(5,14)(6,9)(11,16,12,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,23)(22,31)(24,28)(25,32);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=gh,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105888, #106042, #106196, … |
| maximal normal subgroup of | #105888, #106042, #106196, … |
| maximal quotient of | #160574, #160597, #160598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10584);