Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10508 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #394 · Out(G) = C25 |
| 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 | |
180 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 180 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,8)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,3)(2,8)(4,6)(5,15)(7,14)(9,12)(10,16)(11,13)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,4,5,14,7,15,6,3)(2,9,10,16,12,8,11,13)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)Polycyclic presentation:
2^8|b^2=h,c^2=d,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=egh,f^b=fg,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10509, #240864, #240865, … |
| maximal normal subgroup of | C2×#10509, #240864, #240865, … |
| maximal quotient of | #104903, #104904, #104907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3497);