Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | Aut(G) = C22×#2196 (order 512) · Inn(G) = #2196 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #124 | C23 |
| prime radical | #3289 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,2,8,13)(3,14,17,18)(4,15,9,21)(5,20,23,10)(6,11,25,30)(7,26,28,12)(16,32,31,22)(19,29,27,24);(1,3)(2,9)(4,13)(5,21)(6,24)(7,16)(8,17)(10,18)(11,22)(12,27)(14,20)(15,23)(19,26)(25,29)(28,31)(30,32);(1,4,8,9)(2,3,13,17)(5,22,23,32)(6,18,25,14)(7,27,28,19)(10,29,20,24)(11,15,30,21)(12,31,26,16)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fg,d^b=dh,e^b=efgh,g^b=gh,d^c=df,f^c=fh,g^c=gh,e^d=egh,f^d=fh,f^e=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13680, #512875, #512877, … |
| maximal normal subgroup of | C2×#13680, #512875, #512877, … |
| maximal quotient of | #151145, #151149, #151165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6668);