Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
859 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 859 conjugacy classes of subgroups — too many to draw. Only its 141 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)(3,7)(4,14)(5,17)(6,10)(8,16)(9,19)(11,26)(12,29)(13,22)(15,25)(18,24)(20,28)(21,31)(23,30)(27,32);(1,3,6,13)(2,7,10,22)(4,11,15,27)(5,12,19,31)(8,23,24,20)(9,21,17,29)(14,32,25,26)(16,28,18,30);(1,4)(2,8)(3,11)(5,18)(6,15)(7,20)(9,25)(10,24)(12,30)(13,27)(14,17)(16,19)(21,32)(22,23)(26,29)(28,31);(5,19)(9,17)(12,31)(14,25)(16,18)(21,29)(26,32)(28,30)Polycyclic presentation:
2^8|b^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139741, #139745, #144365, … |
| maximal normal subgroup of | #139741, #139745, #144365, … |
| maximal quotient of | #245064, #245067, #245069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16886);