Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10524 |
| automorphism | Aut(G) = C22×#72231160 (order 4096) · Inn(G) = #394 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
174 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,10,12,7)(4,6,13,8)(11,15,16,14)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(2,6)(3,11)(4,13)(7,10)(8,9)(12,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(1,3,4,11,5,12,13,16)(2,7,8,15,9,10,6,14)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=g,c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fgh,e^c=egh,f^c=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10527, #238932, #238933, … |
| maximal normal subgroup of | C2×#10527, #238932, #238933, … |
| maximal quotient of | #104880, #104883, #104889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3515);