Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
199 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7,10,6,3,5,9)(4,8,14,16,13,11,12,15)(17,18)(19,28,31,39)(20,32)(21,26)(22,36)(23,27)(24,30,40,35)(25,33)(29,44,47,38)(34,41)(37,42)(43,45,48,46);(1,3,6,2)(4,11,13,8)(5,9,7,10)(12,15,14,16)(17,19,22,30,23,31,21,35)(18,24,36,28,27,40,26,39)(20,29,37,45,34,47,33,46)(25,38,32,43,42,44,41,48);(1,4,6,13)(2,8,3,11)(5,14,7,12)(9,16,10,15)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,44)(26,41,36,32)(28,43,39,48)(30,45,35,46)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94784, #94786, C2×#9008, … |
| maximal normal subgroup of | #94784, #94786, C2×#9008, … |
| maximal quotient of | #99538, #99540, #99544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1996);