Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,12)(11,15)(17,18)(19,23)(20,24)(21,22);(2,5)(6,7)(9,10,13,12)(11,16,15,14)(18,21)(19,24);(9,11)(10,14)(12,16)(13,15)(17,19,22,24)(18,23,21,20);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,24)(22,23);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,16)(13,15)(17,21,22,18)(19,20,24,23)Polycyclic presentation:
2^8|b^2=g,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97884, #172689, #172690, … |
| maximal normal subgroup of | #97884, #172689, #172690, … |
| maximal quotient of | #557189, #557225, #557534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38950);