Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(17,18)(19,23)(20,22)(21,24);(2,3)(4,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,21,24,23);(2,3)(6,7)(10,11)(14,15);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|b^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #58260⋊C3, #72341196 |
| maximal subgroup of | #132505, #487356, #487911, … |
| maximal normal subgroup of | #132505, #487356, #487911, … |
| maximal quotient of | #585522, #587635, #589932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51248);