Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = #555 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2420 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,14)(4,18)(5,9)(6,23)(7,12)(8,13)(10,21)(11,17)(15,24)(16,25)(19,26)(20,28)(22,27)(29,30)(31,32);(1,3,6,5,8,17,7,16)(2,9,12,14,13,25,23,11)(4,19,21,15,22,30,20,31)(10,26,18,32,28,29,27,24);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,24)(11,26)(12,27)(13,28)(14,29)(16,30)(17,31)(18,23)(25,32);(1,5,6,17,8,16,7,3)(2,11,12,9,13,14,23,25)(4,19,20,31,22,30,21,15)(10,26,27,24,28,29,18,32)Polycyclic presentation:
2^8|b^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501862, #501863, #504518, … |
| maximal normal subgroup of | #501862, #501863, #504518, … |
| maximal quotient of | #280915, #280916, #280917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17974);