Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72303335 (order 4096) · Inn(G) = C23 · Out(G) = C2×#10994 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C23 |
| commutator, socle | C22 | C2×C4×C8 |
| prime radical | C23 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,35)(29,33)(30,38)(31,36)(32,37)(39,45)(40,43)(41,44)(42,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,27,21,28,30,39,22,29,31,40,32,41,42,47)(18,23,25,34,26,35,38,45,24,33,36,43,37,44,46,48);(3,9)(6,12)(10,15)(13,16)(18,24)(23,33)(25,36)(26,37)(34,43)(35,44)(38,46)(45,48)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,c^a=ch,c^b=cf112 conjugacy classes — view the full character table.
| maximal subgroup of | #128235, #128236, #128237, … |
| maximal normal subgroup of | #128235, #128236, #128237, … |
| maximal quotient of | #126532, #126573, #128215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4400);