Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,23)(13,28)(14,16)(15,21)(17,26)(18,25)(20,35)(22,39)(24,34)(27,36)(29,33)(30,37)(31,40)(32,38);(1,3)(2,5)(4,8)(6,7)(11,20)(15,30)(19,35)(21,37)(22,29)(24,40)(31,34)(33,39);(9,11,14,20)(10,15,16,30)(12,24,27,40)(13,29,25,22)(17,33,32,39)(18,34,28,31)(19,36,35,23)(21,38,37,26);(3,8)(5,7)(10,16)(15,30)(17,32)(18,28)(19,35)(23,36)(31,34)(33,39)Polycyclic presentation:
2^8|a^2=e,c^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33893, #7623713, #7623717, … |
| maximal normal subgroup of | C2×#33893, #7623713, #7623717, … |
| maximal quotient of | #497385, #497546, #497726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26881);