Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| 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 | C42 | C24 |
| prime radical | #3514 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,4,9)(3,8,12,16,13,7,14,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,10)(9,15)(11,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,3)(2,8)(4,14)(5,12)(6,13)(7,10)(9,15)(11,16)(18,23)(22,30)(24,31)(25,32);(1,4,6,5)(2,9,10,11)(3,12,13,14)(7,15,8,16)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,f^2=g,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 | #221324, #221325, #221330, … |
| maximal normal subgroup of | #221324, #221325, #221330, … |
| maximal quotient of | #160632, #160638, #160647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10669);