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 | #3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 635 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:
(1,2,5,10,6,9,18,24)(3,11,14,25,13,7,17,20)(4,8,15,23,16,22,29,32)(12,19,27,30,26,21,28,31);(1,3)(2,7)(4,12)(5,17)(6,13)(8,21)(9,11)(10,25)(14,18)(15,28)(16,26)(19,22)(20,24)(23,30)(27,29)(31,32);(1,4,6,16)(2,8,9,22)(3,12,13,26)(5,15,18,29)(7,19,11,21)(10,23,24,32)(14,27,17,28)(20,30,25,31);(2,9)(5,18)(7,11)(8,22)(14,17)(15,29)(19,21)(27,28)Polycyclic presentation:
2^8|a^2=g,c^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 | #135530, #139744, #139757, … |
| maximal normal subgroup of | #135530, #139744, #139757, … |
| maximal quotient of | #213721, #213724, #213728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14767);