Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3487 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 181 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 — 4 generators on 32 points:
(1,2,6,12,7,11,4,9)(3,13,14,22,16,8,20,24)(5,18,17,26,19,10,21,25)(15,23,30,31,29,27,28,32);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,24)(10,23)(11,13)(12,22)(17,28)(18,27)(19,29)(21,30)(25,32)(26,31);(1,4,7,6)(2,9,11,12)(3,14,16,20)(5,17,19,21)(8,22,13,24)(10,25,18,26)(15,28,29,30)(23,31,27,32);(1,5,7,19)(2,10,11,18)(3,15,16,29)(4,17,6,21)(8,23,13,27)(9,25,12,26)(14,28,20,30)(22,31,24,32)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135531, #139741, #139751, … |
| maximal normal subgroup of | #135531, #139741, #139751, … |
| maximal quotient of | #213723, #213726, #213730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14769);