Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | Aut(G) = #72319052 (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 | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,11,18,28)(3,12,14,30,13,29,31,26)(5,10,20,27,21,8,17,25)(7,22,24,15,23,16,32,19);(1,3,6,13)(2,7)(4,16,18,22)(5,19,21,15)(8,12)(9,26)(10,29)(11,23)(14,17,31,20)(24,25)(27,32)(28,30);(3,13)(7,23)(12,29)(14,31)(15,19)(16,22)(24,32)(26,30)Polycyclic presentation:
2^8|a^2=e,b^2=g,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 | #94774, #152070, #152087, … |
| maximal normal subgroup of | #94774, #152070, #152087, … |
| maximal quotient of | #149583, #149615, #149704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6587);