Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 229376 |
| 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 | |
673 conjugacy classes of subgroups; only the 389 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 389 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(2,6)(4,11)(7,12)(10,15)(18,22)(19,26)(24,28)(27,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fgh,d^2=fg,e^2=g,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 | #53385⋊C7 |
| maximal subgroup of | C2×#53385, #8105934, #8108676, … |
| maximal normal subgroup of | C2×#53385, #8105934, #8108676, … |
| maximal quotient of | #606851, #606853, #606854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46373);