Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator, socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)Polycyclic presentation:
2^8|a^2=f,b^2=de,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| Fitting subgroup of | #12895⋊C3 |
| maximal subgroup of | C2×#12895, #494098, #494099, … |
| maximal normal subgroup of | C2×#12895, #494098, #494099, … |
| maximal quotient of | #135704, #135708, #146809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5883);