Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2021 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3021 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,11)(4,14)(5,18)(8,22)(9,25)(10,15)(12,29)(13,19)(16,28)(20,23)(21,27)(24,32)(26,30);(1,2)(3,8)(4,15)(5,9)(6,10)(7,14)(11,23)(12,30)(13,24)(16,20)(17,25)(18,27)(19,26)(21,31)(22,29)(28,32);(1,3,6,13)(2,8,10,24)(4,16,14,12)(5,19,18,11)(7,20,15,30)(9,26,27,23)(17,28,31,29)(21,32,25,22)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,g^b=gh,d^c=dg,e^c=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13354, #498940, #498941, … |
| maximal normal subgroup of | C2×#13354, #498940, #498941, … |
| maximal quotient of | #142225, #142230, #142272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6342);