Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #401 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,18,25)(3,7,13,21,12,20,17,26)(4,14,16,29,15,8,22,24)(11,19,28,31,27,23,30,32);(1,3)(2,7,9,20)(4,11,15,27)(5,17)(6,12)(8,23)(10,26,25,21)(13,18)(14,19)(16,30,22,28)(24,31)(29,32);(1,4)(2,8,9,14)(3,11,12,27)(5,16)(6,15)(7,19)(10,24,25,29)(13,28,17,30)(18,22)(20,23)(21,31)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11319, #359420, #360371, … |
| maximal normal subgroup of | C2×#11319, #359420, #360371, … |
| maximal quotient of | #121085, #121114, #121115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4307);