Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | Aut(G) = #72284619 (order 2048) · Inn(G) = #379 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(17,18)(19,27,30,23)(20,31,32,42)(21,26)(22,25)(24,33,39,35)(28,44)(29,41,34,38)(36,40)(37,46)(43,47)(45,48);(1,2,6,9)(3,8,11,16)(4,12,5,7)(10,15,13,14)(17,19,22,30)(18,23,25,27)(20,28,32,45)(21,34,36,29)(24,37,39,43)(26,41,40,38)(31,44,42,48)(33,47,35,46);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,39)(26,42)(27,43)(29,47)(30,45)(31,40)(33,36)(34,46)(38,44)(41,48)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94547, C2×#8529, #187863, … |
| maximal normal subgroup of | #94547, C2×#8529, #187863, … |
| maximal quotient of | #97045, #97046, #97047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1517);