Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,24,37)(12,17)(13,27)(14,19)(15,20)(16,23,32,40)(18,29)(22,34,25,38)(26,36)(28,31,39,33)(30,35);(2,5)(3,7)(9,11)(10,16)(12,22)(13,28)(14,25)(15,24)(17,31)(18,34)(19,33)(20,32)(21,35)(23,26)(27,37)(29,40)(30,39)(36,38);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,30,29)(16,31,32,33)(18,27,36,35)(21,34,37,38)(23,39,40,28)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,e^2=fg,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #128904, #128905, #128939, … |
| maximal normal subgroup of | #128904, #128905, #128939, … |
| maximal quotient of | #122856, #122857, #122858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4912);