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 | #3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 493 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:
(1,2,5,9,6,10,21,28)(3,11,14,29,13,25,19,31)(4,15,18,27,17,8,20,26)(7,22,24,16,23,30,32,12);(1,3)(2,7)(4,16)(5,19)(6,13)(8,25)(9,24)(10,23)(11,15)(12,17)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32);(4,17)(8,15)(11,25)(12,16)(18,20)(22,30)(26,27)(29,31)Polycyclic presentation:
2^8|a^2=f,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 | #133840, #135530, #139730, … |
| maximal normal subgroup of | #133840, #135530, #139730, … |
| maximal quotient of | #129222, #129258, #129347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5697);