Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1781 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 476 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,4,9)(3,13,14,31,16,28,22,30)(5,18,17,29,21,10,20,27)(8,23,26,19,25,32,24,15);(1,3)(2,8)(4,14)(5,19)(6,22)(7,16)(9,24)(10,28)(11,26)(12,25)(13,18)(15,21)(17,23)(20,32)(27,30)(29,31);(1,4,7,6)(2,9,12,11)(3,14,16,22)(5,20,21,17)(8,24,25,26)(10,29,18,27)(13,30,28,31)(15,23,19,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133842, #135532, #139738, … |
| maximal normal subgroup of | #133842, #135532, #139738, … |
| maximal quotient of | #129223, #129259, #129346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5698);