Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,27)(19,28,29,45)(20,25,22,37)(21,33,36,47)(24,39,40,31)(26,32,44,35)(30,46,38,43)(34,41,48,42);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,29)(24,40)(28,45)(30,38)(31,39)(34,48)(41,42)(43,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,24)(20,30)(21,34)(22,38)(23,29)(25,41)(26,43)(27,40)(28,33)(31,35)(32,39)(36,48)(37,42)(44,46)(45,47);(2,8)(6,9)(7,16)(14,15)(17,20,23,22)(18,25,27,37)(19,30,29,38)(21,32,36,35)(24,41,40,42)(26,33,44,47)(28,46,45,43)(31,34,39,48)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ceh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139136, #139140, #143563, … |
| maximal normal subgroup of | #139136, #139140, #143563, … |
| maximal quotient of | #496433, #496487, #496740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26791);