Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 688128 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(7,12)(10,15)(14,16)(18,22)(19,25)(24,29)(26,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,6)(4,11)(7,12)(10,15)(18,22)(23,28)(24,29)(30,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53389⋊C3, #53389⋊C7 |
| maximal subgroup of | C2×#53389, #8105943, #8109167, … |
| maximal normal subgroup of | C2×#53389, #8105943, #8109167, … |
| maximal quotient of | #606852, #607491, #607499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46377);