Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #390 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,8)(4,6)(5,7)(9,10)(11,19,22,15)(12,23,26,16)(13,18)(14,17)(20,30)(21,35,27,31)(24,32)(25,36,28,33)(29,34)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,24)(13,27)(14,22)(16,32)(17,19)(18,35)(20,25)(21,29)(23,40)(26,39)(28,38)(30,33)(31,34)(36,37);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,23)(18,36)(19,37)(21,39)(22,38)(24,27)(25,29)(31,40)(32,35)(33,34)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93914, #137427, #137437, … |
| maximal normal subgroup of | #93914, #137427, #137437, … |
| maximal quotient of | #121618, #121619, #121645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4352);