Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| 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×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,14,18)(11,15,13,22)(12,16,21,24)(17,19,23,20);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,18,22)(12,19,21,20)(16,23,24,17);(1,3,5,4)(2,7,6,8)(10,16)(11,13)(12,20)(15,17)(18,24)(19,21)(22,23);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=e,d^2=e,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 | #492001, #492006, #492008, … |
| maximal normal subgroup of | #492001, #492006, #492008, … |
| maximal quotient of | #497669, #497670, #497939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26954);