Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,12)(13,16)(14,19)(15,20)(17,18)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(10,16,20,19)(11,21,22,18)(12,23)(17,24);(1,4,5,3)(2,6,7,8)(9,11,13,21,15,22,14,18)(10,17,19,23,20,24,16,12);(2,7)(6,8)(9,12,14,24,15,23,13,17)(10,18,16,22,20,21,19,11)Polycyclic presentation:
2^8|b^2=eg,c^2=e,d^2=f,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,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151358, #151360, #151476, … |
| maximal normal subgroup of | #151358, #151360, #151476, … |
| maximal quotient of | #326103, #326106, #326108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21928);