Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19)(12,16)(13,26,27,17)(14,18)(15,21,29,36)(20,33)(22,34)(23,37)(24,39,40,31)(25,32)(28,35,30,38);(9,11,14,22)(10,15,18,29)(12,23,25,20)(13,21,27,36)(16,30,32,28)(17,19,26,34)(24,38,40,35)(31,37,39,33);(2,5)(6,8)(9,12,14,25)(10,16,18,32)(11,20,22,23)(13,24,27,40)(15,28,29,30)(17,31,26,39)(19,33,34,37)(21,35,36,38)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94171, #94175, #94488, … |
| maximal normal subgroup of | #94171, #94175, #94488, … |
| maximal quotient of | #124659, #124689, #124717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4492);