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 |
| prime radical | #2206 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 247 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:
(2,5)(6,7)(9,10)(11,20)(12,17)(13,16)(14,27)(15,21)(18,26)(19,25)(22,28)(23,30)(24,32)(29,38)(31,36)(33,40)(34,39)(35,37);(1,2,4,5)(3,7,8,6)(9,11,12,21)(10,15,17,20)(13,22,24,36)(14,23,25,37)(16,31,32,28)(18,29,33,34)(19,30,27,35)(26,39,40,38);(10,16)(11,21)(13,26)(14,25)(15,28)(17,32)(18,19)(20,31)(22,38)(24,40)(27,33)(29,35)(30,34)(36,39);(1,3,4,8)(2,6,5,7)(11,21)(15,20)(22,36)(23,37)(28,31)(29,34)(30,35)(38,39)Polycyclic presentation:
2^8|b^2=e,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,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×#33937, #7623875, #7623877, … |
| maximal normal subgroup of | C2×#33937, #7623875, #7623877, … |
| maximal quotient of | #497616, #497673, #497959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26925);