Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | Aut(G) = #72319013 (order 4096) · Inn(G) = C2×#354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,21,29)(3,12,15,30,14,25,31,28)(4,8,17,26,18,9,20,27)(7,22,23,13,24,19,32,16);(1,3,6,14)(2,7)(4,16,18,13)(5,19,21,22)(8,25)(9,12)(10,28)(11,24)(15,20,31,17)(23,27)(26,32)(29,30);(1,4,6,18)(2,8,11,9)(3,13,14,16)(5,17,21,20)(7,12,24,25)(10,26,29,27)(15,19,31,22)(23,30,32,28)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dg,e^b=efg,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94773, #152075, #152080, … |
| maximal normal subgroup of | #94773, #152075, #152080, … |
| maximal quotient of | #149584, #149616, #149703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6588);