Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,25,21,26,20,30)(19,27,28,32,29,23,31,24);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,31)(21,29)(22,28)(24,25)(26,27)(30,32);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,25)(26,32)(27,30);(2,6)(5,7)(10,13)(14,16)(17,20,21,22)(18,25,26,30)(19,28,29,31)(23,32,27,24)Polycyclic presentation:
2^8|a^2=f,b^2=egh,c^2=g,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142592, #142595, #142596, … |
| maximal normal subgroup of | #142592, #142595, #142596, … |
| maximal quotient of | #362807, #364325, #370125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22298);